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FUNDAMENTALS 
OF HIGH SCHOOL 

MATHEMATICS 



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RUGG-CLARK 



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FUNDAMENTALS 

OF HIGH SCHOOL 

MATHEMATICS 

A TEXTBOOK DESIGNED TO FOLLOW 
ARITHMETIC 

By Harold O. Rugg 

Department of Education 
University of Chicago 

and John R. Clark 

Department of Mathematics 
Chicago Normal College 




YONKERS-ON-HUDSON, NEW YORK 

WORLD BOOK COMPANY 

2126 Prairie Avenue, Chicago 

1919 



60 



WORLD BOOK COMPANY M^ 



THE HOUSE OF APPLIED KNOWLEDGE 



QH 



Established, 1905, by Caspar W. Hodgson ' J x . 0[ 

yonkers-on-hudson, new york 
2126 Prairie Avenue, Chicago 

The purpose of this house is to publish 
books that apply the world's knowledge to 
the world's needs. Public-school texts in 
mathematics should be designed in terms of 
careful studies of social needs as well as in 
terms of methods by which pupils learn. 
This book on general mathematics is the first 
of the Rugg-Clark Mathematics Texts, and 
it will be followed by texts for Junior High 
Schools and for Elementary Schools. Each 
of these texts will embody the result of years 
of scientific investigation and experimental 
teaching 



NOV 17 1919 



rcmt: fhsm-3 



Copyright, 1918, 1919, by World Book Company 

Copyright in Great Britain 

All rights reserved 



©CI.A536592 

'Yvfl I 



<3f 

1 SCIENTIFIC METHOD IN THE CONSTRUCTION 
OF SCHOOL TEXTBOOKS 

The traditional course of study in high school mathematics needs 
to be completely reconstructed. Five years of intensive investi- 

' gation * have established striking inadequacies in the first-year 
course: both what we teach and how we teach it. First: more 

J than half of the conventional first-year course will never be used 
by the vast majority of our pupils. Second : the claims which 
we make that we are training pupils "to think intelligently" 
will be difficult, if not impossible, to substantiate. The very 
content and organization of the course tends to inhibit this. 
Most of it provides little or no opportunity for training in 
"problem-solving." Our courses of study emphasize habit- 
formation and rote memory, and these courses are almost exactly 
determined by the textbook. Finally : standardized tests given 
to more than 100 high schools show clearly that neither in 
securing formal skill nor in developing powers of analytical 
thinking is our instruction satisfactory. 

Striking needs: (1) courses of study based on a clear-cut pro- 
gram ; (2) a real psychology of how children learn mathematics, 
expressed through new types of " wordy " textbooks and manuals 
of method. Courses must be constructed on principles of social 
worth and "thinking" outcomes that have been definitely 
established. Current courses have not been so organized. 
The slate should be wiped clean and a new course organized on 
a clear-cut program. No material should be included which 
cannot be defended either on the basis of social worth, or upon 
the probability of relatively worth-while thought power. Fur- 
thermore, the present course, as it came down from its former 
position in the upper college curriculum, has retained that em- 
phasis upon rigorous, logical organization and upon brief alge^ 

* The evidence which was collected is reported in detail in Scientific 
Method in the Reconstruction of Ninth Grade Mathematics, by H. O. Rugg 
and J. R. Clark ; Department of Education. Bureau of Publications, Uni-;' 
versity of Chicago, 1918. 



iv Scientific Method in Construction of Textbooks 

braic symbolism which characterizes the thinking of " mathema- 
ticians." Children, however, do their thinking in terms of 
detailed word-symbols ; only laboriously do they take on the 
more abbreviated methods of symbolic thinking which are typical 
of mathematical manipulation. The solution is clear : we need 
"wordy" textbooks — textbooks in which the transition from 
thinking in detailed word-symbols to that in abbreviated letter 
or algebraic symbols and in their manipulation is made so 
gradually as to keep, step by step, just ahead of the pupil's 
mental advance. " Gradation " of subject matter must receive 
a real psychological significance in the mind of the textbook 
writer and of the teacher. Basic to all these statements, how- 
ever, is the need for the development of a real psychological 
analysis of how children learn mathematics. 

These facts, coupled with the accumulation of dissatisfaction 
and denunciation on the part of both school administrators and 
lay critics of current mathematical courses, point to the pressing 
need of setting in motion a program for completely reconstruct- 
ing our whole scheme of mathematical instruction. 

"FUNDAMENTALS OF HIGH SCHOOL MATHEMATICS" IS A 
TRANSITION BOOK, INTENDED TO FILL AN IMMEDIATE 
NEED FOR REQUIRED MATHEMATICS IN THE NINTH 
GRADE 

The pressing need is for a complete rebuilding of the material 
of the ninth school year. There is much evidence that the 
11 required " mathematics of the future high school course will 
stop with the ninth year. Hence, the course of study submitted 
herewith is based on the assumption that the mathematics of 
the ninth grade will be the last year required of all children. 
Under the present organization of the seventh- and eighth-grade 
courses, therefore, this new ninth-grade course must include all 
the fundamental mathematical notions and devices which can 
be taught in one school year and to children of that grade of 



Scientific Method in Construction of Textbooks v 

maturity. Furthermore, it is based on the assumption — yes, the 
demand — that there be one year of mathematics beyond arithmetic 
required of all children who remain to our ninth grade. 

Although the immediate need is an acceptable "transition" 
course for the ninth grade, the permanent need is for a scientifically 
constructed unit course of study for the junior high school grades, 
seventh, eighth, and ninth. The authors are conducting investi- 
gations which are basic to the development of such a course of 
study. The course will be published only after thorough scien- 
tific study and experimentation in the classroom. The present 
book, Fundamentals of High School Mathematics, therefore, they 
regard as a "transition" book to fill a pressing need in "first- 
year mathematics," much of the material of which will shortly 
be found distributed with other materials over the eighth and 
ninth school grades. 

"FUNDAMENTALS OF HIGH SCHOOL MATHEMATICS" IS 
CONSTRUCTED ON A DEFINITE SET OF PRINCIPLES 

I. THE SELECTION OF SUBJECT MATTER ON THE CRI- 
TERIA OF SOCIAL WORTH AND "THINKING" OUTCOMES 

It is the judgment of the present writers that a course of 
study should rest upon a definitely founded program of selection 
and arrangement of subject matter. Courses of study should, 
periodically at least, be recast, not by elimination from or 
accretion to the stock course, but rather by spreading out all 
the curriculum materials (mathematical in this case) in full 
view, and taking them into the course in the degree to which 
they satisfy the criteria which have been set up. Doing this in 
mathematics, one includes, first, selected material which is now 
in the traditional first-year course, and second, adds much 
material which is either not taught at all to high school stu- 
dents or is taught to only a limited portion in some advanced 
year. 



vi Scientific Method in Construction of Textbooks 

Two principles should control the selection of subject matter : 
first, social worth — material to be included must prove of social 
worth (in school, in the home, in the occupation, or in leisure 
activities) to a definite portion of our student body ; second, 
"thinking value" — it must be possible to show for material 
whose social utility is doubtful that a thorough grasp of those 
quantitative principles which are needed for sound and com- 
plete thinking in life will be hampered by their exclusion. 

Acting on tKese two criteria, the authors' investigations of 
" how much mathematics " should be experimented with in the 
classroom and finally included in the course, have demanded 
the exclusion of at least 35 per cent of the material of current 
first-year algebra. The time usually given to formal skill in 
handling polynomials (with the four fundamental operations), 
highest common factor, least common multiple, the mastery of 
7 to 17 cases in " special products " and factoring, skill in 
manipulating complicated forms of fractions and fractional 
equations, can have no justification in terms of this criterion of 
social worth. 

Neither does it find a place because of its " thinking " value. 
This is all " formal skill " material, provides no opportunity for 
training in the technique of reasoning, and certainly does not 
contribute to a grasp of " scientific law." 

1. What to include in the new course on the criterion of social 
worth. The course must include, even on the basis of social 
Worth alone, training in (a) the use of letters to represent num- 
bers ; (b) the use of the simple equation ; (c) the construction 
and evaluation of formulas ; (d) the finding of unknown dis- 
tances by means of (1). scale drawings, (2) the principle of 
similarity in triangles, (3) the use of the properties of the right 
triangle (ratios of the sides as expressed in the Hypotenuse 
Rule, and in the cosine and the tangent of an angle) ; (<?) the 
preparation and use of statistical tables and graphs to represent 
and compare quantities (this includes the grasp of elementary 
statistical measures). 



Scientific Method in Construction of Textbooks vii 

2. The application of the "thinking" criterion. So much from 
the social criterion. The writers are of that group, however, 
that regard " thinking" outcomes as coordinate in importance with 
the more commonly accepted " social utility" 

The primary function of mathematical instruction. Their 
thesis in constructing this course of study is this : The central 
element in human thinking is the ability to see relationships 
clearly. In the same way the primary function of a high school 
course in mathematics is to give ability to recognize relationships 
between magnitudes, to represent such relationships economically 
by means of symbols, and to determine such relationships. To 
carry out this aim the course of study, therefore, should be 
organized in such a way as to develop ability in the intelligent use 
of the equation, the formula, methods of graphic representation, and 
the p7-operties of the more important space forms in the expression 
and determination of relationships. 

Note the importance of that last phrase — "in the expression 
and determination of relationships" ; i.e. of "law." The course 
of study in mathematics must not only contain socially worth- 
while material but it must cooperate with other subjects in the 
curriculum in providing training in the development of the sci- 
entific attitude. This, in turn, can come only through constant 
practice in meeting real problem-situations, and in a grasp of 
the principle of "functionality," i.e. of "dependence" or, more 
concretely, of relationship. 

But practice in meeting problem-situations demands that the 
whole course be constructed around a core of problem-solving. 
Unfortunately, algebra courses have been deprived of most of 
their " training " value. An emphasis upon formalism, drill, the 
routine practice in manipulation of meaningless symbols, and 
lack of genuine motive are typical examples of the way in which 
we have hampered teachers in the development of problem- 
solving abilities. The general practice of devoting 80 per cent 
of the exercise-material to these formal drill examples, leaving 



viii Scientific Method in Construction of Textbooks 

only 20 per cent for the verbal problem, — which of all types 
provides most completely opportunity for "thinking," — has 
been radically modified in the course submitted in this book. 
The entire course has been organized around a central core of 
" problem-solving." Even the purely formal materials themselves 
have been so organized, wherever possible, as to provide an 
opportunity for real thinking and not mere habit formation. 

Teaching children to understand and express "law." Our 
courses of study have failed, generally, to the present time, to 
give our high school students a grasp of functionality, i.e. of 
scientific " law," and how to express it. Thus, both the basic 
mathematical purpose of the course and the foundational 
" thinking " purpose of the course have not been fulfilled. 
Hence the importance, in this complete recasting of the course, 
of attempting to build it in such a way as to contribute con- 
stantly to ability in the expression and determination of rela- 
tionship. Chapters VIII and XVI and scattered problem- 
material throughout every chapter provide the type of definite 
training that the writers' experimentation has shown is neces- 
sary and can be given to help bring about the desired outcome 
in a one-year course. 

II. THE ARRANGEMENT OF THE SUBJECT MATTER IN A TEXTBOOK 

The psychological criterion and the principle of mathematical 
sequence control the arrangement. Two principles only should 
control the sequence and gradation of subject matter : (1) math- 
ematical sequence ; (2) learning difficulty. The two principles 
operate to control the organization of the material in this book. 
Two striking illustrations can be given of the application of the 
psychological criterion. The first has to do with the teaching 
of special products and factoring; the second, that of signed 
numbers. A further and basic example can be found in the 
new method of writing a mathematics book — i.e. the " wordy " 
textbook. 



Scientific Method in Construction of Textbooks ix 

The writers have experimented in the classroom with this 
reconstructed course of study. Each has taught first-year 
classes under the critical observation and comment of the 
other. The work has eventuated in an important body of 
material concerning " how children learn mathematics." (This 
material will be presented in a book, The Psychology and Teach 
ing of Junior High School Mathematics, some time during the 
school year 1919-1920.) It has been clearly demonstrated 
that special products and factoring can be satisfactorily taught 
and habituated in 7 class periods, — these contrasted with the 
30 class periods of the common practice of the day. 

Similarly our experimental teaching of the course with signed 
numbers introduced both early and late has supplied what 
appears to the writers, and to more than 50 cooperating high 
schools, to be conclusive evidence for using negative numbers 
only in the second half of the course. This necessitates the 
complete reconstruction of the order and complexity of the 
material of the first half year, and leads to a type of gradation 
that satisfies the basic psychological criterion for arranging the 
subject matter of courses ; namely, the mental content of the 
course must keep just one step in advance of the developing 
content of the pupil's mind. 

A SUMMARY OF DISTINCTIVE CONTRIBUTIONS TO THE. 
TEACHING OF MATHEMATICS ILLUSTRATED BY THE 
CONTENT AND ARRANGEMENT OF "FUNDAMENTALS OF 
HIGH SCHOOL MATHEMATICS" 

A scientific program has been developed for constructing 
courses of study, the steps of which are necessary for the sound 
design of textbooks that will be relatively permanent in our 
generation. It has two striking characteristics : 

(A) The content of the course selected so as to satisfy rigor- 
ous criteria of either social worth or definitely established 
" thinking " value, or both. This procedure is in contrast to 



x Scientific Method in Construction of Textbooks 

that of elimination from, addition to, or rearrangement of a 
stock course of study. In this book it has led to (1) vast 
economy of time by excluding non-essential operations and 
forms; (2) introduction of new material not commonly at- 
tempted or successfully taught in the first-year course : e.g. 
{a) the use of statistical measures, tables, and graphs to repre- 
sent and compare quantities ; (b) the organization of a whole 
course about the central theme of relationship (''functionality") 
and the systematic organization of three methods of representing 
and determining relationship — the graphic method, the tabular 
method, and the equational or formula method ; (c) systematic 
teaching of methods of indirect measurement — i.e. the finding 
of unknown distances by scale drawings, similar triangles, and 
the use of the properties of the right triangle. 

(B) Psychological arrangement of textbooks secured only by 
real classroom experimentation and cooperative teaching by many 
teachers. For the present very general practice of making 
school textbooks at the desk and without careful classroom 
studies of how children learn, the authors substituted a unique 
experimental procedure. Following a year of detailed experi- 
mentation in their own classes the tentative course of study was 
printed, sold at cost by the authors, and taught by teachers of 
very typical experience and training in 62 high schools. From 
these teachers a most searching criticism of the organization of 
the book was obtained. In the light of this detailed analysis 
the book has been completely rebuilt. The writers submit it 
now for general use in the belief that it is constructed in ac- 
cordance with the ways in which children learn and that it fits 
the necessary administrative demands of the classroom. Several 
illustrations can be given of the contributions that can be made 
by this type of experimentation : 

(1) A new type of textbook — a "wordy" textbook — one 
that will teach itself. There is much evidence for the conclusion 
that the exposition of the text develops so gradually in accord- 



Scientific Method in Construction of Textbooks xi 

ance with the way in which children " learn " that the rank and 
file of pupils can read it and work the problems unaided. This 
provides not only a helpful and complete guide for the inex- 
perienced teacher (which is most pertinently needed in all the 
school subjects) but leaves the classroom time of the experienced 
teacher relatively free for the introduction of important supple- 
mentary material. 

(2) The postponement of signed numbers to the second half 
of the course, providing for a very gradual and unique develop- 
ment of the subject matter of the first half. 

(3) The development of a new method of teaching special 
products and factoring which saves at least 20 school days in 
the course. 

(4) Graphic representation is an integral part of the course, 
and is treated throughout as a method of representing quantity 
— not as an isolated operation. 

HOW THE COURSE PREPARES FOR THE STUDY OF 
ADVANCED MATHEMATICS 

A sound course of study in ninth-grade mathematics must 
prepare adequately for conventional third-semester algebra and 
plane and solid geometry and trigonometry which will be taken 
by an increasing number of students. The striking fact is that 
the construction of a text on a scientific program results in a 
course that prepares for such courses even more completely 
than the conventional "first-year algebra." 

The formal material that has been excluded not only has no 
" social " utility outside the school, but it is not used in these 
advanced courses. The new material that has been added is 
direct preparation for them. For example, the three chapters 
that discuss the finding of unknown distances, the chapters on 
expressing and determining relationship, and direct and inverse 
variation, develop intensively concepts and tools which are in- 



xii Scientific Method in Construction of Textbooks 

dispensable to the mastery of geometry, trigonometry, and the 
higher courses. Furthermore, such a procedure gives a far 
more thorough training in the basic algebraic skills. Thus the 
course satisfies the necessary administrative requirement that it 
must fit into an established sequence of school courses. 

THE RELATION OF SCIENTIFIC SCHOOLBOOK CONSTRUC- 
TION TO ENTRANCE REQUIREMENTS OF PRIVATE COL- 
LEGES 

No book can satisfy the criteria of social ivorth and thinking out- 
comes and at the same time satisfy the entrance requirements of 
a small group of private colleges. Fundamentals of High School 
Mathematics does not attempt to do this. The entrance require- 
ments for a small group of private colleges must not be per- 
mitted longer to thwart the sound construction of public 
secondary-school textbooks. For any school which wishes to 
use Fundamentals of High School Matliematics, and also prepare 
for college entrance, the authors are issuing a supplementary 
pamphlet which will contain material needed to do tlie latter but 
which will be of little or no service in further college work or in 
mathematical work outside the school. It is so designed as either 
to fit into the regular course at various places or to be used as 
a review and elaboration of the formal material that is now 
included in the ninth-grade course. 

THE TEACHER'S TESTS FOR A TEXTBOOK 

Textbooks should be selected for use in our schools, which 
will satisfy the following tests : 

(1) Does the subject matter presented in the book sufficiently 
justify itself from the point of view of its use or importance 
either in later school courses, in situations outside the school, 
or in the thinking outcomes obtained from its study? 

(2) Is the subject matter presented in the textbook organized 
in terms of the ways in which children naturally learn ; that is, 



Scientific Method in Construction of Textbooks xiii 

has the psychological criterion been kept constantly and ade- 
quately in mind ? 

(3) Is the pupil who takes the course permitted to do real and 
genuine thinking ? Does he have ample opportunity for prac- 
tice in " problem solving," that is, is the subject matter of the 
course organized primarily around a core of " problem solving " 
situations ? 

The application of these criteria in the construction and selec- 
tion of school textbooks goes far to bring about the type of 
reconstruction for which the writers' investigation shows there 
is a real demand. 

Harold O. Rugg 
John R. Clark 

Chicago, Illinois 
August 2, 1919 



THOSE WHO CONTRIBUTED ESPECIALLY TO 
THE MAKING OF THIS BOOK 

No schoolbook can be made to fit class needs without the coop- 
eration of many teachers and administrators. The superin- 
tendents, principals, and selected teachers of 62 high schools 
made it possible, by their progressive interest and hearty coop- 
eration, to fit the material of the course to the practical needs 
of the typical " first-year " mathematics class. 

Many suggestions for improvement and practical adjustment 
of the material were made by Mr. J. A. Foberg, Crane High 
School and Junior College, Chicago ; Mr. William Betz, East 
High School, Rochester, New York; Mr. L. E. Mensenkamp, 
High School, Freeport, Illinois ; Miss Lillian Barnes, High 
School, Desplaines, Illinois ; Miss Florence Morgan, High 
School, Highland Park, Illinois ; Miss Flora E. Balch, Town- 
ship High School, Evanston, Illinois. These teachers gave of 
their insight and energy unsparingly, and their contribution has 
been very large. 

The initial classroom experimentation of the authors would 
have been impossible without the cooperation of Mr. C. W. 
French, Principal of the Parker High School, Chicago. By his 
interest in the work and his help in arranging class programs, 
the authors' paired-teacher plan of experimental teaching was 
made possible. 



CONTENTS 

CHAPTER PAGE 

I. How to Use Letters to Represent Numbers in 

Solving Problems i 

II. How to Use the Equation 22 

III. How to Construct and Use Algebraic Expres- 

sions -39 

IV. How to Find Unknown Distances by Means of 

Scale Drawings : The First Method . . 54 
V. A Second Method of Finding Unknown Distances : 

The Use of Similar Triangles .... 75 
VI. How to Find Unknowns by Means of the Right 

Triangle 89 

VII. How to Represent and Compare Quantities by 

Means of Statistical Tables and Graphs . 116 
VIII. How to Represent and Determine the Relation- 
ship between Quantities that Change Together 147 
IX. The Use of Positive and Negative Numbers . 166 
X. The Further Use of the Simple Equation . 196 
XI. How to Solve Graphically Equations which Con- 
tain Two Unknowns 229 

XII. How to Solve Equations with Two Unknowns by 

Algebraic Methods 245 

XIII. How to Find Products and Factors . . -257 

XIV. The Use of Fractions with Letters . . . 282 
XV. Literal and Fractional Equations . . . 300 

XVI. How to Show the Way in which One Varying 

Quantity Depends upon Another . . .319 

XVII. Square Roots and Radicals . . . .337 

XVIII. How to Solve Equations of the Second Degree 351 

Index 367 

xv 



FUNDAMENTALS OF 
HIGH SCHOOL MATHEMATICS 



CHAPTER I 

HOW TO USE LETTERS TO' REPRESENT NUMBERS IN 
SOLVING PROBLEMS 

Section 1. It saves time to use abbreviations and let- 
ters, instead of words, to represent numbers. In order to 
save time in reading and writing numbers in your studies 
in arithmetic, you have already found it convenient to use 
certain abbreviations or letters to represent numbers. For 
example, instead of " dozen" you have used " dos." to stand 
for 12; C to stand for 100; M for 1000; cwt. (hundred- 
weight) for 100 lb. ; mo. (month) for 30 days, etc. It is 
necessary that we learn more about this new way of repre- 
senting numbers by letters because we shall use it in all our 
later work in mathematics. 

EXERCISE 1 

PRACTICE IN USING ABBREVIATIONS AND LETTERS TO REPRESENT 
NUMBERS 

1. How many eggs are 6 doz. eggs and 2 doz. eggs ? 

2. How many days in 3 mo. and 2 \ mo. ? 

3. Change 5 yr. and 2 mo. to mo. 

4. If 1 ft. = 12 in., change 4 ft. + 5 in. to in. 

5. If R (ream) stands for 500 sheets of paper, how 
many sheets in 2 R -f- 3 R ? 

6. Change 5 yd. — 2 ft. — 3 in. to in. 



2 Fundamentals of High School Mathematics 

7. Using d for 12, how many eggs in 2 d -f 3 d 
eggs ? 

8. How many sheets of paper in 5 R + & R — 6 R 
sheets ? 

9. Change 5^ + 4 ;?z— 6 m -\- m to smaller time 
units, that is, to " days*," if m = 30 days. 

10. Change ly (yards) -f 6/ (feet) to smaller units, 
that is, to " inches." 

11. If y = 3/, how many/ in 4jj/ -+- 6y — 2y? 

12. If h (hour) equals 60 m (minutes), and m equals 
60 s (seconds), how many s in 2 h + 3 m} 

13. lf.c = 100, what is the value of 2 <: + 4 c + ^ ? 

14. A printer uses M for 1000. How many en- 
velopes are there in 5 M + 3 M — M ? What 
would they cost at 11.50 per M? 

15. How many cents in 4 q -f 6 d, if q and «^ stand 
for the number of cents in a quarter and dime 
respectively ? 

16. If x = 8, what is the value of & x -\-Zx + 1x 
-4*? 

In these examples, you have used abbreviations or single 
letters to represent numbers or known quantities. We 
make use of abbreviations or single letters instead of 
1 because You 

State the reason here 

need a great deal of practice in doing this. The next ex- 
ercise will give more practice in representing numbers by 
letters in different kinds of examples. 



A New Way to Represent Numbers 

EXERCISE 2 
FURTHER PRACTICE IN USING LETTERS FOR NUMBERS 

1. Change 10 y -f 4/ to inches (smaller units) if 
y and f stand for the number of inches in a 
yard and in a foot, respectively. 

2. Express 2 // + 5 m in seconds. 

3. What will 4 R + 3 R + R sheets of paper cost 
at \<f, per sheet? 

4. At 4/ each, what will 1 d — 2d eggs cost? 

5. The length of the rec- ~ r 
tangle in Fig. 1 is 
represented by the 

expression 3/, and the 2 r 

width by the expres- 
sion 2 f. What ex- 
pression will represent 
the perimeter? How 

many inches in the perimeter if/ = 12 inches ? 

6. Change 7/> + 5/ — 8/ -f 3/ to ounces, if p 
stands for the number of ounces in one pound. 

7. The expression 14 y + 8m + 5d represents the 
age of a pupil in your class. Express this 
pupil's age in days or as a certain number of d. 

8. Find the cost of 4| T of coal at 30 cents per 
cwt. y using the relation, T— 20 cwt. 

9. If d— 4 q, find how many q in 3 d+ 5 d. 

10. If y = 12 m, and m = 30d, how many ^/ in 

2^+4^? 



4 Fundamentals of High School Mathematics 

11. A boy earned 27 dollars in a month ; his father 
earned n dollars. How many dollars would 
both earn in 3 months, if n stands for 60 dollars ? 

12. What is the numerical value, of 4 b when b is 5 ? 
When b is 8 ? 

13. The side of a square is represented as s inches 
long. What will represent its perimeter ? 

14. If b represents the number of feet in the base 
of a rectangle, and h represents the number of 
feet in its height, what will represent its perim- 
eter ? What will represent its area ? 

15. If pencils cost c cents each, what will n pencils 
cost ? 

16. What will represent the cost of one tablet if n 
tablets cost c cents ? 

17. If n represents a boy's age, what will represent 
his father's age, if the father is 25 years older 
than the son ? 

18. The width of a rectangle is represented by w. 
If its length is 6 inches longer than its width, 
what will represent the length ? What expres- 
sion will stand for the perimeter ? 

THE PRACTICAL USE OF LETTERS FOR NUMBERS IN 
FORMULAS 

Section 2. Further need for abbreviated language : Short- 
hand rules of computation. People who have found it 
necessary to compute over and over again the areas or 
perhheters of such figures as rectangles, triangles, circles, 
etc., have found it very convenient to abbreviate the rules 
for solving these problems into a kind of shorthand expres- 



A New Way to Represent Numbers 




sion which can be more easily writ- 
ten or spoken than the long rules. 
For example, suppose you wanted 
to make a complete statement, 
either in writing or orally, concern- 
ing how to find the area of the 
rectangle which is represented by 
Fig. 2. You might express it, as 
you did in arithmetic, as follows : 

(1) The number of square units in the area of 
a rectangle is the number of units in its base 
times the number of units in its height. 

This long word rule can be greatly shortened by using 
abbreviations or suggestive letters to represent the number 
of units in each of its dimensions. Thus, a shorter way of 
expressing this rule is : 

(2) Area = base x height. 

A third and still more abbreviated way of expressing it is : 

(3) A = b x h, 

in which A, b, and h mean, respectively, the number of 
units in the area, base, and height. And finally, remem- 
bering that b x h is usually written as bk, the entire state- 
ment becomes : 

(4) A=bh. 

This last statement tells us everything that the first 
statement did, and requires much less time to read or to 
write. Such algebraic expressions are called formulas. 

Section 3. What is a formula? From the previous illus- 
tration we see that a formula is a shorthand, abbreviated 
rule for computing. We must remember, however, that 
the formula A = bh is, at the same time, an equation. 



Fundamentals of High School Mathematics 



HOW TO COMPUTE AREAS AND PERIMETERS BY 
MEANS OF FORMULAS 

EXERCISE 3 
COMPUTATION OF THE AREA OF RECTANGLES BY THE FORMULA 

l. Illustrative example. Find 
the area of a rectangle (Fig. 3) in 
which b = 10 and h = 7.b, using 
the formula 

A = bh. 

Solution : (1)" A = bh. 

(2) A = 10 x 7.5. 

(3) A = 75. 




Find A when b = 8.25 and h = 4. 

Find A when h = 4.5 and b = 12. 

Find b when A = 50 and A = 3. 

What is ^ if £ = 6.5 and A = 5.4 ? 

What is AifA = 40 and £ = 6} ? 

Find i4 if 'A = 2.5 and b = 6.4. 

What is b if A = 450 and // = 22.5 ? 

If A = 200 and £ = 7.5, what does h equal ? 

10. If A — 625 and h = 50, what does b equal ? 

11. What is A if b = 40 and /£ is twice as large as b ? 

12. Find A if h = 16.2 and £ = | of //. 

13. £ = 12and/* = f £. What is ^ ? 

14. Findy2 if // = 20 and h + b = Z2. 

15. Write a formula for finding the base of any rectangle. 

Section 4. Perimeters of rectangles. In Section 3 we 
saw that it was convenient to use a formula for the area 



A New Way to Represent Numbers 



of a rectangle. In the same way 
it is helpful to have a formula for 
the perimeter of any rectangle. 

Since the perimeter of any 
rectangle is the sum of the bases 
and altitudes, the shortest way to 
express this is : 

(1) Perimeter = 2x base plus 2 x height, 
or by the formula 

(2) P = 2 6 + 2/i. 




Fig. 4 



EXERCISE 4 
COMPUTATION OF THE PERIMETERS OF RECTANGLES BY THE FORMULA 

1. Illustrative example. Find the perimeter if the base 
is 13 and the height is 9 ; 
or, more briefly, find P if b = 13 and h = 9. 
Solution: (1) P=2b + 2h. 

(2) P=2-13+2.9. 

(3) P = 26 + 18 = 44. 



2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 



Find P if h = 10.5 and b = 9. 
Find P if h = 18.4 and b = 12.8. 
What is k if P = 40 and b = 10 ? 
What is b if P = 60 and h = 14 ? 
Find h if P = 18.4 and b = 4.6. 
If P = 110 and h = 22.5, what is b ? 
What is P if h = 18 and b = 2 hi 
P = 100. Find b and kiib= h. 
What is AitP = 120 and £ = \P ? 



Section 5. The formula for the area of any triangle. 

What is the area of this triangle if its base is 12 ft. and its 



8 Fundamentals of High School Mathematics 



height is 8 ft. ? How do you find 
the area of any triangle if you 
know its base and altitude ? 

Show that the most economi- 
cal way to state this rule, or re- 
lation, between the area, the 
base, and the height, is by the 

formula A = — . 



The examples in the following exercise will give you prac- 
tice in using this important formula. 




EXERCISE 5 



COMPUTATION OF THE AREA OF TRIANGLES EY THE FORMULA 



l. Illustrative example. 
Find the value of A if b - 22 and h = 12. 
bh 



Solution : (1) A = — 
K J 2 



(2)A = 



22 x 12 264 



= 132. 



Write your work in a neat, systematic form. 

2. Find the value of A if b = 18 and h = 6J. 

3. What is the value of A if b = 12.5 and // = 20? 

4. If h = 16.8 and b = 28, what does A equal ? 

5. What is the value of b if A = 300 and h = 50 ? , 

6. Find // if A = 240 and b = 20. 

7. Determine b if A = 100 and h = 15. 
What is A if // = 6.25 and b = 10.5 ? 
Find the value of A if b = 22 and h = T 6 T b. 
What is b if A = 120 and h = \A ? 
k = 20 and b = | h. What does A ,equal? 



8. 

9. 
10. 
11. 



A New Way to Represent Numbers g 

12. Can you find b and // if A = 100 and b = 2 // ? 

13. Two triangles have equal bases, 10 in. each, but 
the height or altitude of one is twice that of the 
other. Are their areas equal ? Show this by 
an illustration. 

14. What change occurs to A if b is fixed in value, 
but if h gets larger? What is the relation be- 
tween A and // if b is fixed ? 

HOW TO USE LETTERS IN TRANSLATING WORD 
STATEMENTS INTO ALGEBRAIC STATEMENTS 

Section 6. Word statements about quantities may be 
much more briefly expressed by using a single letter to 
represent a quantity. In the first section we saw that it 
saved time to use abbreviations or letters to make short- 
hand rules of computation. Now we shall show that 
entire word statements about quantities may be expressed 
much more briefly by using a single letter to represent a 
number. To illustrate, consider next the four different 
ways of writing the statement of the same example. 

Illustrative example. 

f (a) There is a certain number 
(a) The "word 1 ' method of such that if you add 5 to 

stating the example. it the result will be 18. 

[ What is the number ? 

(6) An abbreviated way to ( ,,. ,__. . , , 1B , 

3 \ (b) What no plus 5 equals 18? 

write it. [ v J 

(c) A more abbreviated way f , \ _. 

* x * ' j (c) No. + 5 = 18. 

to write it. [ v 

(d) The best way to write it. {(d) n + 5 = 18. 

It is clear that in all these cases the number is 13, and 
that it is most easily represented by the single letter n. 



io Fundamentals of High School Mathematics 

Thus, the fourth method, n + 5 = 18, illustrates the very- 
great saving that is obtained by the use of single letters 
for numbers. This method will be used throughout all 
our later work. One of the aims of this course is to help 
you solve problems by better methods than you knew in 
arithmetic. 

EXERCISE 6 

Express the following word statements in the briefest 
possible way, using a single letter to represent the quantity 
you are trying to find. 

Illustrative example. Write, as briefly as possible : A 
certain number increased by 7 gives as a result 16. 
This may be most briefly expressed : 
n + 7 = 16, 
or, n = 9. 

1. There is a certain number such that if you add 
12 to it, the result will be 27. What is the 
number ? 

2. If John had 7 more marbles, he would have 18. 
How many has he ? 

3. If the length of a rectangle were 5 inches less, 
it would be 21 inches long. What is its length ? 

4. A certain number increased by 12 gives as a 
sum 35. What is the number ? 

5. The sum of a certain number and 7 is 18. Find 
the number. 

6. Three times a certain number is 21. Find the 
number. 

Explanation : Again, to save time, we agree that 
3 times n (8 times p, or 12 times x, etc.) shall be 
written 3 • n or, more briefly, 3 n. Understand, 



A New Way to Represent Numbers n 

therefore, that whenever you meet expressions 
like 8£, 12 x, 17 y, etc., they mean multiplication, 
even though no "times" sign (x) is printed. 
In the same way f of a certain number is written 

- n or — ; - of a certain number, - or - n. 
3 3 2 2 2 

7. Two thirds of a certain number is 10. Find the 
number. 

8. Three fourths of a certain number is 15. Find 
the number. 

9. Three fifths of a certain number is 27. What 
is the number ? 

10. The difference between a certain number and 
5 is 9. What is the number ? 

"The difference between two numbers" means 
"the first number minus the second number." 

11. The difference between a certain number and 
12 is 13. What is the number ? 

12. The sum of 16 and a certain number is 29. 
Find the number. 

13. The product of 11 and a certain number is 77. 
Find the number. 

14. If your teacher had $50 less, he would have 
$15. How much has he? 

15. The quotient of a number and 7 is 3. What is 

the number ? 

16. If the area of a rectangle were increased 
12 sq. ft., it would contain 40 sq. ft. What is 
its area ? 



12 Fundamentals of High School Mathematics 

17. 13 exceeds a certain number by 4. What is 
the number ? 

Explanation : Does this mean that 13 is larger, or 
smaller, than the certain number ? How do you 
determine how much larger one number is than 
another ? 

18. Two thirds of the number of pupils in a class 
is 28. How large is the class ? 

19. Tom lacked $7 of having enough to buy a $50 
Liberty Bond. How much did he have ? 

20. Three times a certain number plus twice the 
same number is 90. Find the number. 

21. The difference between 20 and a certain num- 
ber is 4. What is the number ? 

22. The number of pennies Harry has exceeds 30 
by 7. How many has he ? 

23. The sum of two numbers is 40. One of them 
is 27. What is the other ? 

24. The difference between two numbers is 21. 
The larger is 60. What is the smaller ? 

25. The product of two numbers is 95. One is 5. 
What is the other ? 

26. The quotient of two numbers is 13. The di- 
visor is 5. Find the dividend. 

In these exercises you have been changing, or translat- 
ing, from the language of ordinary words into algebraic 
language ; you have been making algebraic statements out 
of word statements. The essential thing in this transla- 
tion is the representation of numbers by letters. You should 
note carefully also that you have begun the practice of 
using a letter to stand for a number which is unknown. 



A New Way to Represent Numbers 13 

HOW TO "SOLVE AN EQUATION" 

Section 7. The most important thing in mathematics: 
the EQUATION. In all the examples in Exercise l> you 
have translated word sentences into algebraic sentences. 
These algebraic sentences are called equations. 

They are called equations because they show that one 
number expression is equal to another number expression. 
For example, you have stated that n + 5 = 18. This 
statement merely expresses equality between the number 
expression n + 5, on the left side of the = sign, and the 
number expression 18, on the right side. 

Furthermore, you have been finding the value of the un- 
known number in each of these equations. From now 
on, instead of saying "find the value of the unknown in 
an equation" we shall say : " solve the equation." For 
example, if you solve the equation 

you " find the value " of s ; namely, s = 4. 
EXERCISE 7 

Solve the following equations : 

1. / + 3=8. This might be written: What no. 

+ 3=8, or ? + 3 = 8. 

2. x — 5 = 10. This might be written : What no. 

_ 5 = 10, or ? - 5 = 10. 

3. 2 11 = 25. This might be written : 2 times ? = 25. 

4. 5 a — 275. This might be written : 5 times 
?=275. 

5. i,r = 7. This might be written: -|- times 
? = 7. 



14 Fundamentals of High School Mathematics 

6. | c — 12. This might be written : f times 
? = 12. 

It is always helpful to think of an equation as 
asking a question. Thus, 5 a + 1 = 16 should be 
thought of as the question : 5 times what number 
plus one gives 16 ? 



7. 


2 b + 1 = 21 


8. 


5^-3 = 2T 


9. 


4;tr = 13 


10. 


12 = 3/ 


11. 


5 + n = 11 


12. 


6 - » = 2 


13. 


2/ + 3/ = 35 


14. 


3*.+ 1=16 


15. 


7 £-2 = 12 


16. 


12^ = 27 


17. 


16 = 5j/-l 


18. 


6/ + 3/ = 27 


19. 


13 = 5j/ 


20. 


21 = 5x+l 



21. 


4j/ + 3 = 20 


22. 


6 + 2^=14 


23. 


27 = 5# + 2 


24. 


}» = 18 


25. 


2 £ - 5 = 11 


26. 


4 * + 5 * = 18 


27. 


S y - 8 = 19 


28. 


10j/ = 25 


29. 


5 * - 1 = 14 


30. 


i + *.=>18 


31. 


| 7 =15 


32. 


1 « + i « = 20 


33. 


3 „ + 1 „ = 30 


34. 


i» + i« = 18 



HOW TO REPRESENT TWO OR MORE UNKNOWN NUM- 
BERS, WHEN THEY HAVE A DEFINITELY KNOWN 
RELATION TO EACH OTHER 

Section 8. In the examples which you have worked in 
preceding lessons, you have had to represent only one 
number in each problem. To illustrate, in Example 9, 
Exercise 6, as in all the other examples solved thus far — 
"the difference between a certain number and 5 is 9." 

Only one number has to be represented. But in most 



A New Way to Represent Numbers 15 

of the examples that you will meet in mathematics you 
will have to represent two or more numbers which have a 
definitely known relation to each other. For example, 
consider this problem : 

Suppose Tom has 5 times as many marbles as John 
has. How many do they both have ? 

It is clear that there are two numbers to be represented ; 
namely, the number that Tom has and the number that 
John has. Furthermore, since there is a definite relation 
between these two numbers, that is, one is 5 times the 
other, it is important to see that each can be represented 
by the use of the same letter. 

If you let n stand for the number John has, what must 
represent the number Tom has ? Since the example states 
that Tom has 5 times as many as John, then Tom must 
have 5 n marbles. In the same way, together they have 
the sum of the two ; namely, n + 5 /z, or 6 n. 

The best way to state this, however, in algebraic lan- 
guage, is to use a set form like the following : 

Let n = the number John has. 
Then 5n = the number Tom has, 
and 5 n + n, or 6 n = the number both have. 

The next exercises will show how two or more unknown 
numbers may be represented by using the same letter, if 
the numbers have a definite relation to each other. 



EXERCISE 8 

1. Harry has four times as many dollars as James 
has. If you let n stand for the number of dol- 
lars James has, what expression will stand for 



1 6 Fundamentals of High School Mathematics 

the number Harry has ? for the number they 
together have ? 

2. The number of inches in the length of a rec- 
tangle is 7 times the number in its width. If n 
stands for the number of inches in its width, 
what will represent the number in its length ? 
in its perimeter ? 

3. An agent sold three times as many books on 
Wednesday as he sold on Tuesday. Represent 
the number sold each day. State algebraically 
that he sold 28 books during both days. 

4. There are twice as many boys as girls in a 
certain algebra class. If there are ;/ girls, how 
many boys are there ? How many pupils ? 
State algebraically that there were 36 pupils in 
the class. Find the number of boys. 

5. On a certain day Fred sold half as many papers 
as his older brother. How can you represent 
the number each sold ? the number both sold ? 

6. During a certain vacation period there were 
three times as many cloudy days as clear days. 
Express the number of each kind of days, and 
the total number of days. If the vacation con- 
sisted of 60 days, how many days of each kind 
were there ? 

7. A rectangle is three times as long as it is wide. 
If it is x feet wide, how long is it ? What is 
its perimeter ? 

8. If one side of a square is s inches long, what is 
the perimeter of the square ? State algebraically 
that the perimeter is 108 inches. 



A New Way to Represent Numbers 17 

9. The sum of three numbers is 60. The first is 
three times the third, and the second is twice 
the third. If n represents the third number, 
what will represent the first ? the second ? their 
sum ? State algebraically that the sum is 60, 
and then find each number. Why do you 
think it was advisable to represent the third 
number by n ? 

10. John sold five times as many papers as Eugene. 
If n represents the number Eugene sold, what 
will represent the number John sold ? What 
expression will represent the difference in the 
number sold? Make a statement showing that 
this expression is 32. 

11. A farmer sold four times as many dollars' 
worth of wheat as of corn. If he received x 
dollars for the corn, what will represent the 
amount he received for both ? 

12. A has 11 dollars. B has three times as many as 
A, and C has as many as both A and B. What 
will represent the number of dollars all three 
together have ? 

13. A horse, carriage, and harness cost $500. The 
carriage cost three times as much as the har- 
ness, and the horse twice as much as the car- 
riage. If you let n represent the number of 
dollars the harness cost, what will represent 
the cost of the carriage ? of the horse ? of all 
together? Make an algebraic statement show- 
ing that all three cost $500. Can you now find 
the cost of each ? 



1 8 Fundamentals of High School Mathematics 

14. A man had 400 acres of corn and wheat, there 
being 7 times as much corn as wheat. Show 
how the number of acres of each could be rep- 
resented by some letter. Make an algebraic 
statement showing that he had 400 acres of both. 

15. The rectangle shown in 
Fig. 6 is three times as 
long as wide. State al- 



gebraically that the pe- FlG - 6 

rimeter is 64 in. What are its dimensions ? 

In the problems just studied you have been considering 
two or more numbers which had a definite relation to 
each other and all of which had to be represented by 
using the same letter. For example, you had to note that 
one number was always a certain number of times another 
one, or was a certain part of another one. In each prob- 
lem you had to decide which of the unknown numbers you 
would represent by that letter. In general it is best to 
represent the SMALLEST of the unknown numbers by n or 
by p or by ANY letter. The other numbers must then be 
represented by using the same letter which you selected to 
represent the first one. 

EXERCISE 9 

Write out the solution of each of the following. Be 
sure to use the complete form illustrated below, in solving 
each example. 

Illustrative example. The larger of two numbers is 7 times the 
smaller. Find each if their sum is 32. 

Let s = smaller no. or 8 s = 32, 

Then 7 s = larger no. or s = 4, the smaller number, 

and 7 s + s = 32, and 7 s = 28, the larger number. 



A New Way to Represent Numbers 19 

1. William and Mary tended a garden, from which 
they cleared §72. What did each receive if it 
was agreed that William should get three times 
as much as Mary? 

2. The perimeter of a rectangle is 48 inches. Find 
the dimensions if the length is 5 times the 
width. 

3. The sum of three numbers is 60. The second 
is twice the first, and the third equals the sum 
of the first and second. Find each. 

4. Divide 848 between two boys so that one shall 
get three times as much as the other. 

5. Twice a certain number exceeds 19 by 5. Find 
the number. 

6. The product of a certain number and 5 is 35. 
Find the number. 

7. A man is twice as old as his son. The sum of 
their ages is 90 years. Find the age of each. 

8. The sum of three numbers is 120. The second 
is twice the first and the third is three times 
the first. Find each. 

9. The perimeter of a certain square is 144 inches. 
Find the length of each side. 

10. The perimeter of a rectangle is 160 inches. It 
is three times as long as it is wide. Find its 
dimensions. 

11. William is three times as old as his brother. The 
sum of their ages is 36 years. How old is each ? 

12. One number is five times another. Their dif- 
ference is 16. Find each. 



2o Fundamentals of High School Mathematics 



X 


X 


X 





13. The sum of three numbers is 14. The second 
is twice the first, and the third is twice the sec- 
ond. Find each number. 

14. One number is eight times another. Their dif- 
ference is 63. Find each. 

15. A rectangle (Fig. 7) 
which is formed by 
placing two equal 
squares together has 
a perimeter of 150 
feet. Find the side 
of each square, and fig. 7 
the area of the rectangle. 

16. Three men, A, B, and C, own 960 acres of land. 
B owns three times as many acres as A, and 
C owns half as many as A and B together. 
How many acres has each ? 

17. John sold half as many thrift stamps as Harry 
sold ; Tom sold as many as both the other boys 
together. Find how many each sold, if all sold 
144 thrift stamps. 



TRANSLATION FROM ALGEBRAIC EXPRESSIONS INTO 
WORD EXPRESSIONS 

Section 9. In the previous work you have translated 
from word statements into algebraic expressions. It is 
also very helpful to translate the algebraic expressions back 
into word expressions. For example, n -f- 4 = 13 is the 
same as the words tatement "the sum of a certain number 
and 4 is 13." In the same way, the algebraic statement 
4j = 26 should be translated as follows : 



A New Way to Represent Numbers 21 

" the product of a certain number and 4 is 26," or 
" four times a certain number equals 26." 

The next exercise will give practice in this important 
process, i.e. translating from algebraic statements into word 
statements. 

EXERCISE 10 

TRANSLATE EACH OF THE FOLLOWING ALGEBRAIC STATEMENTS INTO 
WORD STATEMENTS 



1. 

2. 


y + 4 = 20 

2£ + l = 31 


7. 


f + l = 8 
o 


n. 


¥- 8 


3. 


13 = 2 + y 


8. 


}* + 3 = 8 


12. 


c-4 = 12 


4. 


2^ + 3^ = 55 


9. 


5^ = 18 


13. 


20 - n = 12 


5, 
6. 


j» + l = 12 

n 4- 3 /z = 24 


10. 




14. 


^+^ = 10 
2 3 



SUMMARY OF CHAPTER I 

After studying this chapter you should have clearly in 
mind : 

1. It saves time to represent numbers by letters. 

2. It is very economical to abbreviate commonly used 
rules of computation into formulas which consist 
altogether of letters. 

3. Worded problems may be translated into algebraic 
statements. 

4. Equations are statements that two numbers or two 
algebraic expressions are equal. 

5. Solving equations means finding the value of the 
unknown number or letter in the equation. 

6. Algebraic expressions may be translated into word 
expressions. 



CHAPTER II 

HOW TO USE THE EQUATION 

Section 10. The importance of the equation. Nothing 
else in mathematics is as important as the equation, and the 
power to use it well. It is a tool which people use in stat- 
ing and solving problems in which an unknown quantity 
must be found. In the last chapter we saw that the 
formula, or equation, was used to find unknown quantities, 
sometimes the area, sometimes the perimeter, etc. The 
fact that the equation is used as a means of solving such a 
large number of problems is the reason we shall study it 
very thoroughly in this chapter. 

Section 11. The equation expresses balance of numerical 
values. The equation is used in mathematics for the 
same purpose that the weighing " scale " is used by clerks ; 
that is, to help in finding some value which is unknown. 
The scale represents balance of weights ; similarly the 
equation represents balance of numerical values. To un- 
derstand clearly the principles which are applied in dealing 




How to Use the Equation 23 

with equations, we should consider the scale, as repre- 
sented in Fig. 8. In this case a bag of flour of un- 
known weight, together with a 5-pound weight, balances 
weights which total 25 pounds on the other side of the 
scale. 

Now, if n represents the number of pounds of flour, it 
is clear that the equation 

n + 5 = 25 

represents a balance of numerical values. Obviously, n is 
20, for the clerk would take 5 pounds of weight from each 
side, and still keep a balance of weights. 

This principle, namely, that the same weight may be 
taken from each side without destroying the balance of 
weights, can be applied to the equation 

n + 5 = 25. 

That is, we may subtract 5 from each side of the equation, 
giving another equation, 

n = 20. 

This suggests an important principle that may be used in 
solving equations ; namely, — 

The same number may be subtracted from each side 
of the equation without destroying the equality, or 
balance, of values. 

If you take something from one side of the scale, or of 
the equation, what must you do to the other side ? Why ? 

The fact that the equation expresses the idea of balance 
makes it easy to reason about it, and find out all the things 
that can be done without changing the balance or equality. 
The next exercise suggests this kind 'of study of the equa- 
tion. 



24 Fundamentals of High School Mathematics 

EXERCISE 11 

By thinking of the equation as a balance, you should be 
able to complete the following statements. Fill in the 
blanks with the proper words. 

1. Any number may be subtracted from one side 

of an equation if ! is ! from the 

other side. 

2. Any number may be added to one side of an 
equation if • is • to the other. 

3. One side of the equation may be multiplied by 
any number, if the other side is ! , by the 



4. One side of an equation may be divided by any 
number, if the other side is ! by the 



These are very important principles, and are used ia 
solving any equation. They are generally called axioms. 
They must be understood and mastered. The examples 
of the next exercise have been planned to help you leara 
how to apply them. 

EXERCISE 12 

In each of the following examples you can use one of 
the four principles stated above to explain what has 
been done, or to state the reason for doing it. Thus, if 
2jt=8, then ^r = 4, because of the principle: 

" One side of an equation may be divided by a number 
if the other side is divided by the same number." 

For each example,* you are to state the principle which 
permits or justifies the conclusion. 



How to Use the Equation 



25 



5. 



6. 



1. If 4 b = 22, then what is done to each side to 
give b = 5 J ? 

2. If ^_y = 7, then to get y = 14, what do you do 
to each side ? 

3. If x -f 4 = 13, then to get x = 9, what do you do 
to each side ? 

4. If 5*: = 32.5, then what is done to each side to 

give c = 6.5 ? 

If 6 a = 12, then what is done to each side to 

give 3 a = 6 ? 

If y — 4 = 7, then what is done to each side to 

give y = W1 

Hint : Which expression is the larger, y or y — 4 ? 
How much larger ? Then what must be done to 
the smaller expression to make it equal to the 
larger expression ? 

7. If x — 6 = 10, what is done to each side to get x=16? 

8. It is known that y —5 = 7. Why does y — 12 ? 

What is done to n — 6 = 11 to get n = 17 ? 

If a — 1 = 9, then why does a = 10 ? 

If b = 2/i, then what is done to each side to 
give 3 b = 6 /^ ? 

12. If 4 x-r- 3 = 23, then what is done to each side 
to give 4^ = 20? 

13. If 5 y — 3 = 27, then what is done to each side 
to give 5 y = 30 ? 

14. If x + 7 = 19, then to make ;r = 12, what is done 
to each side ? 

15. If 2 c - 4 = 8, then to make 2^ = 12, what is done 
to each side ? 



9 
10 
11 



26 Fundamentals of High School Mathematics 



18. 



19. 



16. If 3 b + 1 = 22, then what is done to each side 
to give b = 7 ? 

17. If 5 £ + 2 = 47, why does 5 b = 45 ? Then why 
does ^ = 9 ? 

If 6 * + 2 = x + 22, then why does 5 * + 2 = 22 ? 
then why does 5x = 20 ? then why does .r = 4 ? 
If you know that 4 w + 3 = w + 27, then why 
does 4 ze; = w + 24 ? then why does 3 w = 24 ? 
then why does w — 8 ? 

These examples are given to emphasize the fact that 
there are certain changes that can be made on both sides 
of an equation, without destroying the balance or equality. 
It should be clear that there must be some axiom or prin- 
ciple to justify every change that is made. 



EXERCISE 13 



Find the value of the unknown number in each of the 
following equations, telling exactly what yon do to each 
side of the equation. 



1. 


.r+5 = 13 


2. 


3^ = 17 


3. 


26 = 4ty 


4. 


2^ + 1 = 19 


5. 


y -5 = 12 


6. 


1^ = 4.5 


7. 


H 


8. 


2^-3 = 17 


9. 


15 = ^ + 7 



10. 


2* + 3* = 85 


li. 


5y + ±y+y = 30 


12. 


5^-2 = 38 


13. 


27=6^-3 


14. 


4^+7 = 47 


15. 


4j/-j/ = 21 


16. 


5* + 1=23 


17. 


21^ = 15 


18. 


J* = 18 


19. 


2 b + 3 b = 42 



How to Use the Equation 



27 



20. <r-4 = 13 

21. 2 £-1 = 18 

22. 18 +* = 13 + 10 



23. 7 + 2^ = 23 + 10 

24. \x+\x = 18 



HOW TO CHECK THE ACCURACY OF THE SOLUTION OF 
AN EQUATION 

Section 12. When is an equation solved? We have 
already noted that an equation is solved when the numerical 
value of the unknown number is found. Thus, the equa- 
tion 4 a + 3 = 29 is solved when the numerical value of a 
is found which makes both sides equal. This leads to 
another very important question ; that is : How can you be 
certain your solution is correct ? In other words, how can 
you test or check the accuracy of your work ? 

For example, suppose that in solving the equation 
4^ + 3 = 29 

one member of your class obtains 8 for the value of a. Is his 
result correct ? There is only one way to be sure. That is 
to substitute or " put in " 8 in place of a in the equation, 
to see whether the numerical value of the left side equals 
the numerical value of the right side. In other words, does 
4 . 8 + 3 = 29 ? 

Clearly, not. Therefore, the solution is incorrect ; it does 
not check. Then what is the correct value of a ? Some 
of you doubtless think it is 6|-. Let us test or check by 
substituting 6J for a, to see if the numerical value of one 
side of the equation will equal the numerical value of the 
other side. Does 

4.6|- + 3 = 29? 

Yes. Then the equation is solved, or, to use the more gen- 
eral term, the equation is satisfied when a = 6 J. 



28 Fundamentals of High School Mathematics 

Summing up, then, an equation is solved when a value 
of the unknown is found which satisfies the equation ; that 
is, one which makes the numerical value of one side equal 
to the numerical value of the other side. The solution of 
the equation is checked, or proved, by substituting for the 
unknown number the value which we think it has. If, as 
the result of the substitution, we get a balance of values, 
then we know that the equation has been solved correctly. 

EXERCISE 14 
PRACTICE IN CHECKING THE SOLUTION OF EQUATIONS 

1. The pupils in a class tried to solve the equation 

6^-3 = 39. 
A few decided that a = 7, while the others in- 
sisted that a = 6. Which group was right ? Show 
how they could have checked or tested their re- 
sult. Why, do you think, did some pupils get 6 
for the value of a ? 

2. Does x — 5 in the equation VI x — 7 = 10jr+3? 
In other words, does x = 5 satisfy this equation ? 

3. Would you give full credit on an examination to 
a pupil who said that y — 4 \ would satisfy the 
equation Sy — 4 = 6y + 3 ? Justify your answer. 

4. Show whether the equation b 2 -f 5 b = 24 is satis- 
fied or solved if b = 3 ; if b = 2. 

5. Do you agree that the value of x is 6 in the equa- 
tion 10 x — 4 = 58 ? Justify your answer. 

6. Is the equation — \ h 6 = - b + 8 satisfied 

when b = 8 ? 

7. Does x— 24 satisfy the equation 



How to Use the Equation 29 

8. State in words how the solution of an equation 
is tested or checked. 

9. What is the value of learning to check very care- 
fully every kind of work you do ? 

EXERCISE 15 

Solve each of these equations. Write out your work for 
each one in the complete form illustrated in the first example. 
Check each one so that you can be absolutely certain that 
your work is correct. 

1. Illustrative example. 

6 b - 4 = 24. 

(1) By adding 4 to each side, we get 

6 b = 28. 

(2) By dividing each side by 6, we get 

& = ¥■ 

(3) Checking, 

6 • ^ — 4 = 24. 
28 - 4 = 24. 



2. 5^-2 = 38 * + 1 = j 

3. 6 £ + 3 = 45 5 

,„ n 14. 10£ + 3 = 7£ + 15 

4. I x — x + 6i) 

15. 12^-2 = 5^+26 

5. 10a-3a = m _ rt 

2 16. 13a/ = 2r + 3y+4j/+8 

**-** + » 17. 8 /+2 = 3/+37 

7. 2^ + 1 = 26 ia 4 + 9jr-2jr + 46 , 

8. * + 5* = 20 + * 19 T £_ 3 = 2 ^ + 12 

9. 4^ = 13+^ 2a S w-5 = w + 51 

10. 3^+2^ + 6^=66 21 ^ + 2^+3^ = 48 

11. 5 <: + 3 = 78 22. j + 2j/ + 3_r + 1 = 61 
12 - ^7 = 8 i 23. 5^-2 = 34-^- 



30 Fundamentals of High School Mathematics 



HOW TO GET RID OF FRACTIONS IN AN EQUATION 

Section 13. The use of the most convenient multiplier.* 

In many equations that you have solved already it has been 
necessary to multiply each side of the equation by some 
number. For example, in %x = 10, it is necessary to mul- 
tiply each side by 2, which gives x = 20. Or, if you wanted 
to solve the equation ^x — Z, it is necessary to multiply 
each side by 5, giving x = 15, 

But, suppose you had an equation 

would you get rid of both fractions by multiplying each side 
by 2 ? Would you get rid of both fractions by multiplying 
each side by 5 ? Here, as in all equations of this kind, 
you have to find some number which is a multiple of the 
different denominators. For this reason, in this example, 
10 is the most convenient number by which to multiply each 
term in the equation. 

Illustrative example. 

(1) £* + £*= 14. 
Multiplying each side of (1) by 10, we get 

(2) 10 • \ x + 10 • \x = 10 • 14, 

(3) or 5x + 2x = 140, 

(4) or 7*= 140, 

(5) or x = 20. 

Checking, by substituting the value of x in the original equation, 
gives 

\ . 20 + i • 20 = 14, 
10 + 4 = 14. 

The study of this example shows that we can get rid of 
fractions in an equation if we multiply each side by the 

* This helpful phrase was first suggested to the authors by Mr. J. A. 
Foberg, Crane High School and Junior College, Chicago. 



How to Use the Equation 



3i 



lowest common multiple of the denominators. We shall 
call this the most convenient multiplier. 



EXERCISE 16 
PRACTICE IN SOLVING FRACTIONAL EQUATIONS 

Solve and cJieck each example. 

1. Illustrative example, f n + \ n = 22. what is n ? 

(1) Multiplying each side by the most convenient multiplier, 12, 

gives 

12 • f n + 12 • \ n = 12 • 22, 

or, 8 n + 3 n = 264, 

or, 11 n = 264. 

(2) By dividing each side by 11, we get 

n = 24. 

(3) Checking, by substituting the value of n (24), in the original 
equation, 

I • 24 -f 1 • 24 = 22. 
16 + 6 = 22. 

2. What is the value of x in the equation 

\x+ \x = \\>. 

3. A man spent \ of his income for rent and \ for 
groceries. Using n to represent his income, 
make an equation which will state that he spent 
% 660 for rent and groceries. Solve the equation. 

4. The dimensions of a rectangle are indicated 
on Fig. 9. What equation will state that the 
perimeter is 36 in. ? Solve the equation for /. 




Fig. 9 



32 Fundamentals of High School Mathematics 

5. |^ + ^ = i^ + 13. What is b} 

6. \x + x = \x+\k. 

7. If three fourths of a- certain number be dimin- 
ished by one half of the number, the remainder 
is 10. Find the number. 

8. \ x +\x- 1 = 3. 

10. Three boys together had 65 cents. Tom had 
half as much as Harry, and Bill had two thirds 
as much as Harry. Translate this into an equa- 
tion and solve. 

11. | + 2 + £ = 4 + ~. What does n equal ? 

L o A 

12. ^-£-2 = 14. 

5 

13. One half of a certain number increased by four 
fifths of the same number gives 52 as a result. 
Find the number. 

14. Harry made two thirds as much money last year 
selling the Saturday Evening Post as John 
made ; Edward made three fourths as much as 
John. How much did each boy earn if all 
together earned $ 145 ? 

15. jr+|^-6 = 24. 

16. The sum of the third, fourth, and sixth parts of 
a number is 18. Find the number. 

17. \y-1=\y. is. \b=\b+1. 

19. Twelve increased by one half of a certain number 
gives the same result as fifteen increased by one 
fifth of the certain number. Find the number. 



How to Use the Equation 



33 



20. \n — \n + i ;z = 5< 



25. 



5 = 



21. 


^=^- + 2 
3 4 




• 26. 


n 15 

2 6 


22. 
23. 


\P + £/=/ + 5 

8 = 1^+2 




27. 


7_T 

5 2 


24. 


J + f7 = ^ + 32 




28. 


!/ = ! 




29. 


11 

4 : 


~2 





30. Two thirds of the length of a rectangle is 8 
more than its width. Its perimeter is 64 inches. 
What are its dimensions ? 

31. Two thirds of a certain number is 16 more than 
two fifths of the same number. What is the 
number ? 



32. 


5 — 2 ~ 

y — -jgX 


34. 


M = 2. 




X 


35. 


M=8 




y 


36. 


63_ T 




X 


37. 


2/ 




36 „ 



38. 



33. 



\y 



-U 



2. (Multiply each side by x.) 



2y 



= 4 



39. 



40. 



41. 



42. 



50 _ 


= 10 


.r 




60 


= 3 


144 
3/ 


= 12 


100 

3j 


= 10 



Section 14. How word problems are solved by equations. 
It is important to note the principal steps involved in 
solving word problems. Let us take, as an illustration, 
Example No. 10 in Exercise 16. 



34 Fundamentals of High School Mathematics 

Three boys together had 65 cents. Tom had half as 
much as Harry, and Bill had two thirds as much as Harry. 
How much had each ? 

1. The first important step in solving a word 
problem is to get in mind very clearly what is 
known and to recognize what is to be found 
out. In all problems some things are known 
and some things are to be determined. Thus, 
in this problem, we know how much money all 
the boys have together ; and we also know 
that Tom has half as much as Harry ; further- 
more, we know that Bill has two thirds as 
much as Harry. That is, we see that the state- 
ment of the amount that Tom and Bill each 
has depends upon the statement of the amount 
that Harry has. 

2. But we do not know how much Harry has. 
Then, as in all word problems, we represent by 
some letter, such as n, the number of dollars 
Harry has. In other words, the second step is 
to get clearly in mind what quantities are un- 
known, and to represent one of them by some 
letter. 

3. Next, all the parts or conditions of the prob- 
lem should be expressed by using the SAME 
letter. Thus, if Harry has n dollars, the 
number that Tom and Bill each has should be 
represented by using the same letter n, and 
not some other letter. That is, the word 
statement must be translated into an algebraic 
statement. It is always necessary, and usually 
difficult, to see that there must be a balance, an 



How to Use the Equation 35 

equality, between the parts of the problem. 
Thus, we must see that Harry's money, n, 
plus Tom's money, \n, plus Bill's money, \n y 
must balance, or equal, 65 cents. This gives 
the complete algebraic statement : 
n-\-\n-\-\n — 65. 

4. The equation which we have obtained must be 
solved. A value of the unknown must be found 
which will SATISFY the equation. In this case 
n proves to be 30. 

5. Finally, the ACCURACY of the result must be 
TESTED by substituting the obtained value of 
n in the original word statement of the problem, 
to see if the statement holds true. 

The word statement says that Tom has half as much 
money as Harry. Our solution says Harry has 30 cents. 
Then Tom has 15 cents. Similarly Bill has two thirds as 
much, or 20 cents. They together had 65 cents, or 30 
cents +20 cents+15 cents. Thus our solution checks with 
the word statements in the problem. 

EXERCISE 17 

Translate into algebraic language, and solve each of 
the following word statements. Check each one. 

1. Six more than twice a certain number is equal 
to 12. Find the number. 

2. Four times a certain number is equal to 35 di- 
minished by the number. What is the number? 

3. I am thinking of some number. If I treble it, 
and add 11, my result will be 32. What number 
have I in mind ? 



36 Fundamentals of High School Mathematics 

4. If fourteen times a certain number is dimin- 
ished by 2, the result will be 40. Find the 
number. 

5. What is the value of y in the equation 

4^/ + £^ + 2 = 28? 

6. If seven times a certain number is decreased by 
8, the result is the same as if twice the number 
were increased by 32. Find the number. 

7. An algebra cost 12 cents more than a reader. 
Find the cost of each if both cost $1.64. 

8. The sum of the ages of a father and his son is 
57 years. What is the age of each if the father 
is 29 years older than the son ? 

9. The length of a school desk top exceeds its 
width by 10 inches; and the perimeter of the 
top is 84 inches. What are its dimensions? 

10. Divide $93 between A, B, and C, so that A 
gets twice as much as C, and B gets $10 more 
than C. 

11. Should a teacher give James full credit for the 
solution of the equation 

41^-7 = 3^ + 5 
if he obtained x = ^-? Justify your answer. 

12. Make a drawing of a rectangle whose perimeter 
is represented by the expression 6j + 20, writ-, 
ing the dimensions on the drawing. 

13. The length of a rectangle is 5 inches more than 
twice its width ; its perimeter is 46 inches. 
What are its dimensions ? 



How to Use the Equation 37 

14. A school garden was 3| times as long as wide. 
To walk around it required 31 steps (27 in. 
each). Tell how to find its width, but do not 
actually find it. 

15. Three men went into business together. A put 
in f500 more than B, and C put in one fifth as 
much as B. How much did each put in if they 
together put in $4900? 

16. The smaller of two numbers is 8 more than one 
fifth of the larger. Find each number if their 
sum is 56. 

17. The perimeter of a triangle is 90. The longest 
side is twice as long as the shortest, and the 
other one is three halves as long as the shortest. 
Find the length of each side. 

18. The sum of four numbers is 70. The first one 
is one half of the third; the second is three 
fifths of the third ; and the fourth one is 5 more 
than the third. Find each number. 

19. A farmer sold a certain number of hogs, ;z, at 
$ 20 apiece. What did they bring him ? Then 
he sold twice as many sheep at $14 apiece. 
What did the sheep bring him ? If all together 
brought him $576, how many of each did he 
sell ? 

20. A man had a square piece of ground fenced in 
for a garden. He made it 10 feet longer, and 
8 feet wider. He then needed 236 feet of fence 
to inclose.it. Find its original size. 



38 Fundamentals of High School Mathematics 

SUMMARY OF CHAPTER II 

From your study of this chapter, the following principles 
and methods should be kept clearly in mind: 

1. Equations express balance of value. 

2. If any change is made on one side of the equa- 
tion, the same change must be made on the other 
side. 

3. An equation is solved when a value of the un- 
known is found which satisfies the equation. 

4. The accuracy of your solution is checked by 
substituting the value of the unknown in the 
equation, and noting whether the numerical 
values of the two sides balance. 

5. You can get rid of fractions in an equation by 
multiplying each side by the lowest common 
multiple of the denominators; that is, by the 
most convenient multiplier. 

6. Five important steps must be mastered in solving 
word problems: 

(a) getting clearly in mind what is known and 
what is unknown ; 

(b) representing one of the unknowns by some 
letter ; 

(c) expressing in an equation all parts of the 
problem by using the same letter ; . 

(d) solving the equation ; 

(e) checking the result by substituting the value 
obtained for n in the original word statement. 



CHAPTER III 

HOW TO CONSTRUCT AND USE ALGEBRAIC 
EXPRESSIONS 

Section 15. Meaning of an algebraic expression. We 

have already seen that letters are often used to represent 
numbers. Any symbol, such as a letter, or a group of 
symbols which represents a number, is called an algebraic 
expression. For example, ab is an algebraic expression ; 
c + d is an algebraic expression; 2g + p + y is another; 
2 x + 3y + z, another, etc. 

Section 16. The numerical value of an algebraic expres- 
sion. If we give numerical values to the single letters, then 
the whole expression has a numerical value. Thus, if a = 2, 
and b = 5, then the numerical value of the expression ab is 10 
(that is, 2 x 5). If c — 3, and af=4, then the numerical value 
of the expression c -j- d is 7. Similarly the numerical value 
of other expressions such as 2 g + p —y, depends upon the 
values that we give to^-, /, and y. In this chapter we shall 
learn how to construct and use algebraic expressions. 

EXERCISE 18 

PRACTICE m FINDING THE NUMERICAL VALUE OF QUANTITIES IN 
PRACTICAL FORMULAS 

1. In the formula P = 2b + 2/z find the value of P 
when b — 7.5 and h = 4.2. 

2. Find the value of A in the formula A = bh when 
b = 6.4 and h = 4J. 

3. What is the value of A in the formula A = 

^ifb = 5.8 and 7^ = 4.6? 

4. Determine what the value of V is in the 
formula V= Iwk, if /= 12, w = 10, and h == 9. 

39 



40 Fundamentals of High School Mathematics 

I. "EVALUATION": HOW TO FIND THE NUMERICAL 
VALUE OF AN ALGEBRAIC EXPRESSION 

Section 17. In the examples which you have just solved 
we have used the long expression " What is the value of " 
or "Find the value of" in referring to the particular letter 
which was to be found. Instead of these long expressions 
we shall now use the single word evaluate. It means 
exactly the same thing as the longer expression. Thus 
to evaluate an algebraic expression means to find its 
numerical value, exactly as in the previous examples. 
This is done by " putting in" or by substituting numerical 
values for the letters. A few examples will make this 
clear. 



EXERCISE 19 
SUBSTITUTING NUMBERS FOR LETTERS IN COMMONLY USED FORMULAS 

1. Evaluate A = ^ if b = 10 and h = 14.6. 

2. Evaluate, or find the value of, P in the expres- 
sion P = 2 b + 2 h if b = 26 and h = 12.4. 

3. Evaluate C = 2 irR if R = 14. tt = 3\jL. 

4. Evaluate V — Iwh if /= 10, w = 6 J, and h = 5. 

5. Find the value of i in the formula i—prt if 
/ = 1640, r= T ^, and /= 4. 

6. Evaluate c = - if E = 110 and R = 10.5. 

7. What is /z in the algebraic expression P — 2b 
+ 2/i if ^=80 and ^ = 12.8? 



Constructing and Using Algebraic Expressions 41 



exercise 20 

PRACTICE IN FINDING THE NUMERICAL VALUE OF ALGEBRAIC 
EXPRESSIONS 

In each of the following examples, let a = 2, b = 3, 
c — |, x = 5, and y = 0. 

Illustrative example. 

2 JC 

Evaluate the expression 3 a + 2 c 

Putting in place of each letter its numerical value, gives 

3.2 + 2-i-^ 

2 3 

or 6 + 1 - Z\ 
or3|. 

1. ±a + 5 b-2x 6 x+2y 



2. ab + 6c + y 



20 c 



3. 3 ac -+- 2 for „ a b 

5. t±£ 8. ^ + tffc 

II. THE USE OF EXPONENTS TO INDICATE MULTIPLI- 
CATION IN ALGEBRAIC EXPRESSIONS 

Section 18. Need of short ways to indicate multiplica- 
tion. A very large part of our work in mathematics is 
that of finding numerical values. In many of our problems, 
therefore, we shall need short ways of indicating multipli- 
cation. For example, in arithmetic, the multiplication of 
5 x 5 is sometimes written as 5 2 ; or the multiplication of 
6x6x6 as 6 3 . In algebra, to save time, this notation, 
or method of indicating multiplication, is always used. 
Thus, instead of writing b x b or n x n x 11 we will write 



42 Fundamentals of High School Mathematics 

b 2 or n s . This little number that is placed to the right 
of and above another number tells how many times that 
number is to be used as a factor. These numbers are 
called exponents. Numbers with exponents are read as 
follows : 



-3W 



T 

i 



Fig. 10 



3 a 2 means 3 times a times a, and is read "3a square. 1 ' 1 
This does NOT mean 3 a times 3 a. The exponent affects only 
the a. 

5 ft 3 means 5 times b times b times b, and is read "56 cube." 
This does NOT mean 5 b times 5 b times 5 b. The exponent 
affects only the b. 

Here, as well as throughout all later mathematical work, 
you will need to be able to evaluate algebraic expressions 
which involve exponents. For 
example, the area of the rec- 
tangle shown here is the expres- 
sion 3 W 2 , which is obtained by 
multiplying 3 W by W. Now 
the numerical value of this area 
depends upon the value of W ; that is, if W is 4, then the 
area is 3 • 4 • 4, or 48 ; but if W is 2, then the area is 
3-2-2, or 12. In the same way 
the volume of the rectangular box in 
Fig. 11 is represented by the ex- 
pression 2 X s , or 2x-x*x. Again, 
you see that the numerical value of 
the volume depends upon the value 
of x. Thus, if x is 5, the volume is 
obtained by evaluating the expression 2x s , which gives 
2-5-5-5, or 250. The next exercise gives practice 
in evaluating algebraic expressions which contain ex- 
ponents. 




Constructing and Using Algebraic Expressions 43 



exercise 21 

PRACTICE IN THE EVALUATION OF ALGEBRAIC EXPRESSIONS WHICH 
CONTAIN EXPONENTS 

l. Illustrative example. 

Evaluate 2 ab 2 + 3 a 2 b + ac, if a = 4, b = 3, and c = 1. 
Solution : 2- 4- 3- 3 + 3-4- 4- 3 + 4-1 = 72 + 144 + 4 

= 220. 
Note that the numbers are substituted for, or put in place of, 
the letters. 

Using the values of a, b, and c given in Example 1, 
evaluate each of the following expressions: 



2. 


rt 2 + £2 + ^2 


3. 


Sabc 


4. 


a 2 b + ab 2 


5. 


ac 2 + cb 2 + ba 2 


6. 


a 3 + b* + c 3 


7. 


b a b 


8. 


a— b + c 


a +- b — c 


9. 


a 2 bc 2 


0. 


M+ 1 

a o c 





2« 2 


12. 


# 6 +• b a + <;° 


13. 


# 2 +• be + tf £ 2 


14. 


tf&: 2 -+- tf^V +- #■*&■ 


15. 


tf 2 £v 


16. 


' b 2 + 2bc + c 2 


17. 


b c b 



18. «? + £ + £? 

£ 2 % 2 tf 2 

19. 2^+2A+2^ 2 

20. The formula d = 16 1 2 tells what distance an 
object will fall in any number of seconds. Find 
how far a body will fall in 1 sec. of time, that is, 
when t = 1. Do you believe it ? How could 
you test it ? 

21. Using the formula in Example 20, find how far 
an object will fall in 2 sec. of time, that is, when 
/ = 2. How could you test the truth of this? 



44 Fundamentals of High School Mathematics 

22. The horsepower of an automobile is given by 
the following formula : , in which D rep- 

resents the diameter of the piston, and N the 
number of cylinders. What is the horsepower 
of a Ford, which has 4 cylinders, and in which 
D = 3J in. ? 



III. THE CONSTRUCTION OF FORMULAS 

Section 19. It is very important to be able to make a 
formula for any computation that must be performed over 
and over again. For example, we often have to find the 
area of a square. Instead of saying or writing each time 

"the area of a square is equal to the square of 
the number of units in one of its sides," 

it saves time to use the formula A = s 2 , in which A = 
area and s = one of the sides. This formula tells all that 
the word rule says and requires much less effort. To give 
practice in this kind of work, construct a formula for each 
of the examples in the following exercise. 

EXERCISE 22 

1. (a) Find the volume of a rectangular box 

whose dimensions are 12, 8, and 6 inches. 
What did you do to the 12, 8, and 6 to find 
the volume ? If /, w, and h stand for the 
dimensions of any box, what would you do 
to them to find the volume ? 
(b) Write the formula for the volume of any box. 

2. (a) What is the area of a circle whose radius is 

9 in.? 



Constructing and Using Algebraic Expressions 45 

(b) What did you have to do to the 9 to get the 
area? If the radius is represented by R, 
what would you have to do to get the area? 
Write a formula for the area of any circle. 

3. (a) Draw, or imagine, a cube each edge of 

which is 8 inches. Find the entire surface 
of all the faces of the cube. 
{b) What did you do in (a) to find the entire 
surface ? Just what would you do if each 
edge were s units Jong ? 
Write the formula for the entire surface of 
any cube. " 

4. (a) What is the interest on $400 for 2 years, 

at 6 % ? 
(&) Just how did you solve (a)? How would you 
find the interest on p dollars for t years 
at r % ? 

Write a formula for the interest on any 
principal for any rate and for any time. 

5. (a) How many cubic inches in a block 2' by 3' 

by 4'? 
(b) Make a formula for the number of cubic 
inches in any rectangular solid whose dimen- 
sions are expressed in feet. 

6. Make a formula for, or an equation which tells, 
the cost of any number of pounds of beans at 
12 cents per pound. 

7. What equation or formula will represent the 
area of any rectangle whose base is 5 inches, but 
whose height is unknown ? Evaluate your for- 
mula for h = 3.4. 



46 Fundamentals of High School Mathematics 

8. An automobilist travels 20 miles per hour. 
How far does he go in 2 hours ? in 5 hours ? 
What do you do'to the number of hours to get 
the distance ? If time is / instead of 5, what 
must you do to the t to get the distance ? 
What formula or equation will represent the 
distance he travels in / hours ? Evaluate this 
formula for 

t = 5 hr. 20 min. 



9. You learned in arithmetic how to 

2 3 
product of two fractions ; thus - x - = 

v 5 7 



find the 
2»3 

5-7* 



What is the rule for finding the product of any 
two fractions ? Express this rule as a formula,. 

using - and - for any two fractions. 
b d 

10. In arithmetic you learned how to divide one 

, ■' ' ,, 35373- 7 

fraction by another : e.g. - r -s-- = T x- = - — =■. 
J <5 47454-5 

Using - and - for the fractions, state the rule 
b d 

for division of fractions as a formula. 

11. Solve each of the following fractional equa- 
tions : 



{a) $* = 85 


W 2^ = 4 


(6) |» = 12 

Mir = ia> 


(rtg-e 


<o i-A 


«£■" 



12. Make a formula for each of A, b, and h y in 
which A is the area of a triangle, and b and h 
are the base and height, respectively. 



Constructing and Using Algebraic Expressions 47 

13. Using the formula i=prt, find what principal 
will yield $60 interest in 3 years at 4%. 

14. Determine the value of V in the formula V= Iwh, 
\il— 5.4, w = 4.8 and k = \w, 

HOW TO USE THE TIMED PRACTICE EXERCISES 

On all timed exercises start all pupils working at the 
same moment ; stop them exactly at the end of the allowed 
time. (Set your watch exactly at some even hour, as at 
9 o'clock and minutes.) 

The teacher will read the correct answers and the pupils 
should correct their own work. Pupils should then record, 
on the record sheet on page 49, or on their individual card, 
the number of examples attempted and the number right, 
and compare their rights with the " standard." The authors 
secured the best results by giving a given "timed" prac- 
tice exercise every third or fourth day until pupils reached 
the standard ; after that, once in two weeks will be found 
sufficient to hold the skill. 

Pupils must not write answers on the text. For reasons 
obvious to the teacher it is desirable to vary the example 
with which to begin on a given trial. Occasionally it will 
help to begin with the last example and work back toward 
the first. 

It is very important to keep a record of one's practice 
by trials. Progress in learning is graphed in Chapter VII. 

PRACTICE EXERCISE A (TIMED) * 

The examples in this exercise have been worked by 
pupils in many high schools. On the average they 

* These and subsequent Practice Exercises are from the authors' 
Standardized Practice Exercises, which are now used in 300 cities. They 



48 Fundamentals of High School Mathematics 

succeeded in doing 10 examples right in 4 minutes on 
the fourth or fifth trial. Can you do this on your fifth 
trial ? 



1. 


If C = 


= 4 and/= 


= 2 what does 2 c 2 — 3 cf equal ? 


2. 


Ua = 


= 3 and b = 


= 2 what does 3 ab + <z# 2 equal ? 


3. 


lix = 


= 3 and y- 


= 4 what does xy 1 — Ixy equal ? 


4. 


If c = 


5 and d = 


= 2 what does — H — — equal ? 
5 4 


5. 


lir = 


2 and s = 


4 what does r* + 3r 2 s equal ? 


6. 


lid = 


= 3 and e - 


= 4 what does 4 </ 2 — 2 dfe equal ? 


7. 


If m - 


= 2 and n 


= 3 what does 2 mn 4- ttztz 2 equal ? 


8. 


Ua = 


-- 4 and b = 


= 5 what does <2$ 2 — £ab equal ? 


9. 


Ux = 


= 4 and y - 


= 3 what does — H — -£- equal ? 

L 


10. 


lip = 


- 3 and q - 


= 5 what does / 2 + 2p 2 q equal ? 


11. 


Ha = 


= 3 and b -- 


= 2 what does 3 « 2 — 2 <?# equal ? 


12. 


lir = 


■■ 4 and s = 


= 2 what does 2 rs 4- rc 2 equal ? 


13. 


If * = 


= 3 and v - 


= 4 what does ?^ 2 — 3 uv equal ? 


14. 


Ux = 


- 5 and y - 


x 3 jiti/ 
= 2 what does — H — -f- equal ? 
5 6 


15. 


If b = 


- 3 and c = 


= 2 what does b 2 + 3 £% equal ? 



are distributed by H. O. Rugg, School of Education, University of Chicago, 
printed on manilla cards. Record cards accompany them. Many teachers 
prefer to keep the pupils' record cards. If so, each pupil may draw up on 
cardboard a record card similar to that on page 49. 



Constructing and Using Algebraic Expressions 49 



i 

pq 

% 

|5 
Ba 

WQ 

A<0 

p£ 
WO 

gift 


73 

E 

O 

& 


.3 




































H 


JO 

3 






































ft 








































2 






































3 






































Pi 






































•a 

6 








































r^O' 

p 










































































.2 

+-> 



3 
& 






































3 






































V 
+J 

Q 






































r— 4 
CO 

5 




£ 

* 






































& 

% 













































































wC5 


^ 


K) 





q 


kl 


^ 





* 


N 


^ 


k 


H 


* 


* 





a. 


Of 


ft! 



50 Fundamentals of High School Mathematics 

EXERCISE 23 
PRACTICE IN MAKING ALGEBRAIC EXPRESSIONS 

1. A and B together have $ 30. A has n dollars ; 
how many has B ? 

2. A man can do a certain piece of work in 10 
days. What part can he do in 1 day? in 4 
days ? in x days ? in 2y days ? 

3. John's age is now 30 years. What expression 
will represent his age in 5 years ? in x years 
from now ? n years ago? 

4. A newsboy received 40 cents for n papers. 
What did he receive for each ? for 10 ? 

5. The difference between two numbers is d. If 
n is the smaller one, what is the larger ? What 
expression will stand for the sum of the two 
numbers ? 

6. A rectangular field is x rods long and y rods 
wide. What is the area of the field in square 
rods ? in acres ? What will stand for the value 
of the field if it is worth n dollars per acre ? 

7. A lot is x feet wide and 60 feet long. What 
will stand for its perimeter ? its area ? 

8. How long will it take a man to ride n miles if 
he rides 12 miles an hour ? 

9. Express algebraically the sum of a and b ; the 
difference between a and b; the product of a 
and b ; and the quotient of a and b. 

10. James sold a motorcycle for 1175, thereby 
gaining x dollars. What did it cost him ? 



Constructing and Using Algebraic Expressions 51 

11. Two men, A and B, are 30 miles apart. They 
start to walk toward each other, A at the rate 
of 3 miles per hour and B at the rate of 2 miles 
per hour. How fast do they approach each 
other ? In how many hours will they meet ? 

12. If, in Example 11, A and B had been x miles 
apart, in how many hours would they have met ? 

13. What are consecutive numbers ? If 10 is the 
smaller of two consecutive numbers, what repre- 
sents the larger ? 

14. If n is the middle of three consecutive numbers, 
what represents the other two numbers ? 

REVIEW EXERCISE 24 

1. What does an equation express ? Is 7 + 4 
= 6 + 6 an equation ? 

2. Does 2 ;/ + 1 = 21, if n = 9, make an equation? 
if n = 10 ? 

3. The formula for the perimeter of a rectangle, 
fl = 2a + 2b, contains three tmknown numbers. 
How many of them must be known in order to 
use this formula to solve an example ? 

4. Read each of the following equations as ques- 
tions, and find the value of the unknown number: 

0) 4j + 3 = 21 {d) Zy - 5 = 16 

(b) 20 = 6 + 2^ ( e ) x + x=m 

(c) 5c + 2 = 42 (/) b+ b + 1 = 23 

5. Three times a certain number, plus 2, equals 
38. Find the number. 



52 Fundamentals of High School Mathematics 

6. Donald saved twice as much money as his 
older brother. Express in algebraic language 
that both together saved $ 96. How much did 
each save? 

7. Evaluate the formula V — Z 3 ( V— the volume 
of a cube), if /= 4J. 

8. The first of three numbers is twice the second, 
and the third is twice the first. Find each 
number if their sum is 105. 

9. Construct a formula for the cost of any number 
of eggs at 30 cents per dozen. 

10. What is the difference in meaning between 10 n 
and n 4- 10 ? Does 4 w mean the same as 4 + w ? 



SUMMARY OF CHAPTER III 

The most important principles and methods which we 
have learned in this chapter are the following : 

1. A formula is merely a shorthand rule of 
computation. 

2. Algebraic expressions are "evaluated" or 
"solved" by substituting numbers for the 
letters in the expression. 

3. We have frequent need to be skillful in substi- 
tuting numbers for letters in practical formulas. 

4. Exponents are used as short methods of indicating 
multiplication. An exponent of a number tells 
how many times that number is taken as a factor. 

5. We should construct a formula for any kind of 
problem which we have to solve frequently. 



Constructing and Using Algebraic Expressions 53 

REVIEW EXERCISE 25 

1. Make a formula for the number of revolutions 
made by the front wheel of a Ford car in going 
a mile, if the radius of the wheel is 14 inches. 

2. In what sense does the equation lb — 5 = £ + 25 
ask a question ? 

3. Give one illustration of the advantage of using 
letters for quantities. 

4. What are the four fundamental principles or 
axioms which are used in solving equations? 

5. Does x = 4 satisfy the equation x 2 — 3 x = 6 ? 

6. Using m, s, and d for minuend, subtrahend, and 
difference, respectively, what equation or equa- 
tions can you make from them ? 

7. An autoist travels at an average rate of 24 mi. 
per hour. What distance will he cover in 2 hr. ? 
in 5 hr. ? in 10 hr. ? Make an equation or for- 
mula for the distance he will travel in t hr. 

8. Write a formula for the cost of any number of 
pounds of bacon at 45 cents per pound. 

9. Draw rectangles with bases of 2 inches each, 
but with different heights. What is done to the 
height to get the area ? Then, if h represents 
the height, what must be done to it to get the 
area ? 

What formula will represent the area of any 
such rectangle if h represents the height ? 
10. How do you get rid of fractions in an equation ? 
What is the most convenient multiplier in any 
particular equation ? 



CHAPTER IV 

HOW TO FIND UNKNOWN DISTANCES BY MEANS OF 
SCALE DRAWINGS: THE FIRST METHOD 

Section 20. We need to know how to find unknown dis- 
tances. The methods of mathematics are really all planned 
to help us find unknown values. The equation, which we 
have studied so carefully, is the best algebraic tool with 
which to do that. Many times, however, in practical life 
work the unknown values that we need to know are dis- 
tances. For example, the surveyor may need to know the 
distance across a river and may not be able actually to 
measure it. Or, he may need to know the distance be- 
tween two points, with some other intervening object 
between which prevents him from measuring it directly. 
Now, mathematics has given us three ways to find such an 
unknown distance. In Chapters IV, V, and VI we shall 
discuss these methods. 

The first method is to make a scale drawing, which will 
include in some way the unknown distance. Next, there- 
fore, we shall study how to determine unknown distances by 
means of scale drawings. Before we take up that particular 
subject, however, we must study how to measure the lines 
and angles which make up scale drawings. 

Section 21. The measurement of lines. We are already 
familiar with certain methods of measuring distances. For 
example, we have measured the length of lines, such as the 
distance from A to B or from C to D. If we use a metric 
scale, in which the units are centimeters, the distance from 



B 



Fig. 13 
54 



Finding Unknown Distances by Scale Drawings 55 

A to B, which is read " line AB," is 5.08 centimeters long, 
and the line CD is 6.35 centimeters long. If we use a foot 
rule in which the units are inches, the distance between A 
and B, or the line AB, is 2 inches, and line CD is 2.5 
inches. Note here that the distances or lengths that we 
obtain for these lines depend upon the kind of scale, or 
kind of unity that is used in measuring. 



THE MEASUREMENT OF ANGLES 

Section 22. An angle is determined by one line turning 
about a point in another line. In order to construct scale 
drawings, we must know how to measure angles. Let us 
think of an angle as being formed by one line turning, or 
rotating, about a fixed point on some fixed or stationary line. 
The line O Y turns or rotates about point O. For example, 

•Y 




Fig. U 

in Fig. 14, think of AX as a fixed, or stationary, line. (It is 
easiest always to take this line as horizontal.} Think also 
of another line, say O Y, as turning, or rotating, about some 
point on the fixed line AX, say point O. As the line OY 
rotates about the point O, it constantly forms a larger and 
larger angle with the fixed line AX. (The symbol for 
angle is Z.) The point O, about which the line turns, is 
always the point at which the two sides of the angle meet, 
and is called the vertex of the angle. 



56 Fundamentals of High School Mathematics 

The arrow is drawn to indicate that the line OY is turn- 
ing, or rotating, about the point O. 

Section 23. The unit of angular measurement. Just as 
we have units and scales for measuring straight lines, so 
we have units and scales for measuring angles. Evidently 
the unit with which we must measure the size of the angle 
is one that will measure the amount that the line has rotated 
about the fixed point. Figure 15 shows that we can 




Fig. 15 



think of the rotating line as turning clear around until 
it occupies its original position again. That is, any 
point P on the line OX has turned through a complete 
circle in rotating about O and returning to its original 
position. 

This suggests that the unit with which we measure 
angles will be some definite fraction of the circle. For a 
long time people have agreed that the circle be divided 
into 360 units and that each one of these units of angular 
measure be called a degree. The symbol used for 
degree is a small ° placed at the right above the number. 
For example: 45° is read "45 degree's." Thus, Figs. 16, 
17, and 18 illustrate angles of different sizes or of differ- 
ent numbers of degrees. 



Finding Unknown Distances by Scale Drawings 57 




Fig. 18 

Section 24. The protractor : How to measure 
angles. Just as we use foot rules, yardsticks, meter 
sticks, etc., to measure straight-line distances, so we have 
an instrument called a protractor to measure angular 
distances. Figure 19 shows that the circular edge of the 




Fig. 19. A protractor for constructing and measuring angles. 



58 Fundamentals of High School Mathematics 

protractor is marked off {i.e. is " graduated ") into degrees. 
Note from the figure that the protractor is divided into 
180 equal parts (half of the total number of angular units 
in the circle), called degrees. Sometimes the whole circle 
is used and marked off to give 360°. 




Fig. 20 



The next figure, Fig. 20, shows how to measure an angle 
with a protractor. First, lay the straight edge of the pro- 
tractor so that it will fall exactly upon one of the two 
lines that form the angle, and with the center of the pro- 
tractor exactly upon the vertex, O, of the angle. Then 
the other side of the angle, OB, for example, will appear 
to cut across the circular edge of the protractor. Now 
count the number of degrees from the point where the 
curved edge of the protractor touches OA to the point 
where it crosses the line OB. Hence, in Fig. 20, the 
angle A OB contains 54°. It is very important for us to 
be able to read angles accurately. The next exercise will 
give you practice in reading angles. 



F hiding Unknown Distances by Scale Drawings 59 



EXERCISE 26 
PRACTICE IN MEASURING ANGLES 

l. Measure each of these angles with a protractor 
in the way described in the last paragraph. 




3. 



Fig. 23 

Compare angle A and angle C. Which has the 
longer sides ? What effect has the length of a 
side of an angle upon the size of the angle ? 
Measure each angle of triangle ABC. From the 
results of your measurement, what is the sum 
of all three angles of this triangle ? 

C 




Fig. 24 



60 Fundamentals of High School Mathematics 

XX 7Y 




Fig. 25 



4. How large isZ^r? How many degrees in Z.y> 
in Z z ? How many degrees in the sum of the 
angles of this triangle, X YZ ? 



5. Draw with the protractor an angle of 30 c 
60°; 100°. 



45 c 



6. At each end of a line 3 in. long draw angles of 
50°. Produce these lines until they meet, and 
measure the angle formed by them. How many 
degrees in it ? Compare the lengths of the lines 
you drew. How many degrees does the sum of 
the three angles of this triangle make ? 

7. Draw a triangle 
such as triangle 
ABC, so that AB 
= 4 inches, angle 
^=60° and AC 
= 3 inches. Then 
find the number 
of degrees in an- 
gle B and angle C 




Finding Unknown Distances by Scale Drawings 61 



8. Construct triangle ABC 
so that AC = 4 in., angle 
6^=40°, and CB = 4 in. 
Compare angle A with 
angle B. How many de- 
grees in each? 

Explanation : A triangle hav- 
ing two sides equal, such as 
AC and CB, is an isosceles tri- 
angle. It is proved in geom- Fig. 27 
etry that the angles opposite 

these equal sides are always equal; for example, angle 
A = angle B. How many degrees ought there to be in either 
angle A or angle B? 

Section 25. How to describe an angle. An angle is 
described by using three letters, i.e. the letter which repre- 
sents the vertex is written between the two letters at 





Fig. 28 



the ends of the sides. Thus, Z 1, in Fig. 28, is read as 
angle A OB or angle BOA, and is written Z A OB or 
Z BOA. In the same way, Z 2 is read angle BOC or 
angle COB, and is written Z BOC or Z COB. 



62 Fundamentals of High School Mathematics 



EXERCISE 27 
PRACTICE IN READING ANGLES 

1. Why would it not be clear to read Z 2 as Z. O ? 

2. Read the angle formed by lines OA and OB. 

3. Read the angle formed by lines OB and OC. 

4. Determine the number of degrees in /-AOB> in 
Fig. 29, without using the protractor. 




Fig. 30 



5. If in Fig. 30 you 
know that angle 
ABC is 40° and 
that Z BCA is 90°, 
could you find 
ZCAB without 
measuring it? 
How ? How large is it ? 

Section 26. The sum of the angles of any triangle is 

180°. You have measured the angles of several triangles 
and found that the sum is very close to 180°. If your 
measurements had been absolutely correct, you would 
have obtained exactly 180° for the sum of the angles of 
each triangle. In geometry it is proved without measure- 
ment, that the sum is always 180°. You will use this fact 
in solving the examples in the next exercise. 



Finding Unknown Distances by Scale Drawings 63 



EXERCISE 28 
PRACTICE IN FINDING THE VALUES OF ANGLES IN A TRIANGLE 

1. The number of degrees in each of the three 
angles of a triangle is represented by x> x + 10, 
and 2,r, respectively. Find the size of each 
angle. 

2. In triangle ABC, Z C is equal to Z B, and Z A 
is equal to the sum of Z B and Z C. Find the 
number of degrees in each angle. 



3. 



6. 




Fig. 31 

The first angle of a triangle contains x degrees ; 
the second angle is 15° larger than the first, and 
the third is as large as the sum of the other two. 
Find each angle. 

Angle A in a certain triangle is twice as large as 
Z B } and Z C^is 10° larger than the sum of Z A 
and Z B. How large is each angle ? 

One angle of an isosceles triangle is 50°. Find 
the size of the angles opposite the equal sides. 
See Example 8, page 61. 

Find the angles of a triangle if the first is one 
half of the second, and the third is three fourths 
of the second. 




64 Fundamentals of High School Mathematics 

7. Find the angles of a triangle in which the second 
angle is 10° larger than the first, and the third 
is 10° larger than the sum of the first and second. 

8. Angle B is 10° less than Z A ; Z C is twice as 
large as Z B. How large is each angle ? 

Section 27. We must be able to find unknown distances 
which cannot be measured directly. The preceding sec- 
tion took up only examples in which the distances, linear 
and angular, could be measured directly, by means of 
instruments. There are many instances, however, in 
which the lengths of the lines and the sizes of the angles 
cannot be measured directly. For example, consider the 
case of finding the distance across a river, or the height of 
a tree, which we mentioned at the beginning of the chapter. 
In cases like this we need indirect methods of measuring. 
Mathematics makes it possible for us to determine the 
lengths of such lines by measuring the lengths of other 
lines and the sizes of angles that are related to them. This 
leads us to the main topic of this chapter. 



HOW TO FIND UNKNOWN DISTANCES BY MEANS OF 
SCALE DRAWINGS 

Section 28. How to draw distances to scale. One of the 

methods that you will use commonly in indirect measure- 
ment is that of drawing distances "to scale." So much 
use is made of mechanical drawings that we need to be 
very proficient in making them and in reading them. 
Let us take a simple illustration of the drawing of 
distances "to scale" and of measuring distances on scale 
drawings. 



Finding Unknown Distances by Scale Drawings 65 

Illustrative example. A man starts at a given point and 
walks 2.5 miles east, then 2.5 miles north. How far is he from 
his starting point ? 

First, set point O, in Fig. 32, as his starting point. East is meas- 
ured to the right of point and north above point 0. 



































s 


cat 


ei 


//_ 


It 


nil 


^ 










\ 


( + 




























/> 


























k 


























> 


d 
^ 


• > 


























<9 


V >" 


























^ 

-> 

*' , 


























t 


^ 


> > 


























rf 


>- > 












10 












/ 


¥\ 


y 














N 














y j 
















if! 










s e ! 


i j 


















£ 






,o 


A K 


/ 


























> 


/ 


























X 


























V 































B 












n,a 


St 


2. 


5- 


ill. 


.es 























































Fig. 32 



Second, to represent distances "to scale," we need to select a 
unit of distance on the scale which will represent a unit of distance 
in the example. Let us take, for example, an inch on the draw- 
ing to represent each mile which the man actually walks. 
This is indicated on the scale drawing (Fig. 32) by writing 
'•Scale = 1 in. to 1 mi." It is very important to select the. 



66 Fundamentals of High School Mathematics 

scale unit carefully and always to indicate the scale that has 
been used on the drawing. 

Third, to represent the man's path, we lay off OB horizontally 
to the right of 0, 2.5 miles (on the drawing this amounts to 
2\ inches) and BC vertically, 2.5 miles. Then, by using the 
cross-section paper as a scale, we can measure at once the distance, 
OC, that the man is from his starting point. The distance is 
3.54 inches on the figure, or 3.54 miles. 

This work illustrates by a very simple example how we 
make scale drawings. Mechanical drawings made " to 
scale" are used very commonly by such workers as archi- 
tects, carpenters, machinists, and engineers. 



EXERCISE 29 

PRACTICE IN FINDING UNKNOWN DISTANCES BY THE CONSTRUCTION OF 
SCALE DRAWINGS 

1. Draw to the scale 1 cm. to 2 ft. a floor plan of 
a room 28 ft. by 20 ft. By measuring the 
distance diagonally across the plan, compute 
the diagonal of the room. 

2. Draw a plan of a baseball diamond 90 ft. square 
and find the distance from first base to third 
base. Use 1 cm. to represent 20 ft. 

3. Two bicyclists start from the same point. One 
rides 12 miles north and then 8 miles east ; 
the other rides 10 miles south and then 
6 miles west. How far apart will they be ? 
Use the scale 1 cm. to 2 mi. 

4. Draw to the scale 1 cm. to 4 ft. a plan of the 
end of a garage such as in Fig. 33. Find the 
height from the floor to the top of the roof. 



Finding Unknown Distances by Scale Drawings 67 




Fig. 33 



5. A surveyor sometimes finds it necessary to 
measure the distance across a swamp, such as 
AB in Fig. 34. He measures from a stake 




Fig. 34 



A to a stake C, 120 ft. From C to B he finds 
it is 100 ft. Find, by a scale drawing, the dis- 
tance AB across the swamp, if angle C is 85°. 



!f 



68 Fundamentals of High School Mathematics 

6. How could a surveyor find the distance from 
A to B, if there were some obstacle in the way 



I 



•^ 



Fig. 35 



B 



7. 



preventing his measuring directly the distance 
AB? Could he represent AB as a line in a 
triangle from which he could make a scale 
drawing ? 

Find by a scale drawing the distance AC across 




the river, if it is known that angle A = 80°, 
AB = 200 ft., and angle B = 70°. 

8. Illustrative example. A boy wishes to determine the 
height of a flagpole. A scale drawing will aid him in 
doing this. For example, he can measure a line of any 
length on the ground out from the base of the flagpole. 
Suppose he takes a line 80 feet long. Then he sets at E an 
instrument called a transit, with which he can read the 
angle between the horizontal base line and the line of sight 
from E, where he stands, to H, the top of the pole. He 



Finding Unknown Distances by Scale Drawings 69 



1 

c 
p 



B 



£ 



\ 



\ 



\ 



\ 



\ 



X 



\ 



\ 



X 



known Z. 



X 



\ Known Z.\ ,-. 

\ 1 \k 



< -80' Known 

Fig. 37 



knows also that the angle B is a right angle. So he 
knows the length of the line EB, the size of the angle E and 
the angle B. He constructs a scale drawing to represent 
the known length and the known angles. From this 
drawing he is able to " scale " or measure the height of the 
flagpole. 



The angle BEH or angle E between the horizontal and 
the line of sight in this example is called the angle of 
elevation. If the boy had taken a longer base line, what 
would have been true of the size of the angle of elevation 
with respect to what it was before ? 

9. If the angle of elevation in Fig. 37 is 40° when 
the observer is 80 ft. from the foot of the pole, 
what would be the height of the pole ? 

10. A flagpole 50 ft. high casts a shadow 60 ft. 
long on level ground. What is the angle of 
elevation of the sun ? If the length of a shadow 
cast by this pole increases, what conclusion can 
be drawn concerning the angle of elevation ? 



70 Fundamentals of High School Mathematics 



11. The angle of elevation of the top of a tree is 
42° when the observer stands 30 yd. from the 
tree. How high is the tree ? If the distance 
from the observer to the tree decreases, what 
change in the angle of elevation follows ? 

12. An anchored balloonist from a height HT, 
Fig. 38, of 2500 yd., observes the enemy at D. 



BALLOON 




unknown h 

Fig. 38 

He wishes to compute the distance DT, on 
level ground. To do so, he measures the angle 
which is formed by the horizontal line HL and 
the line of sight HD. This angle is called the 
angle of depression, and has the same number 
of degrees as the angle of elevation. (Can 
you see that this is true ?) He next finds angle 
THD by subtracting the angle of depression 
from 90°, angle THL. He then knows enough 
about the triangle DTH to make a scale draw- 
ing of it. Find DT if the angle of depression 
is 35°. 



Finding Unknown Distances by Scale Drawings 71 



13. From the top of a lighthouse 80 ft. high the 
angle of depression of a ship is 35°. How far 
is the ship from the base of the lighthouse ? 
Compare your result with that obtained by the 
other members of the class. 

14. From the top of a cliff 120 ft. above the sur- 
face of the water the angle of depression of a 
boat is 20°. How far is it from the top of the 
cliff to the boat ? 

15. An observer is 200 ft. from the ground. The 
angle of depression of a point A is 24°, of a 
point B 42°, and of a point C 15°. Which 
point is closest to the observer ? farthest from 
the observer ? 

16. In Fig. 39 measure (1) the angle of elevation 
of point D from point E ; (2) the angle of de- 
pression of point E from 

point D. Compare these 
angles. Note that line 
DH is parallel to EC, 
and that these angles are 
formed by line DE cut- 
ting (or intersecting) 
these two parallels. In 
geometry it is proved 
that such angles are al- 
ways equal. 

17. Is the scale drawing a very accurate method of 
determining unknown distances ? Do many of 
the pupils in the class get the same answer for 
any particular example ? Why ? 




Fig. 39 



72 Fundamentals of High School Mathematics 

SUMMARY OF CHAPTER IV 

The most important things you have learned in this 
chapter are the following : 

1. It is possible to find unknown distances by mak- 
ing a " scale drawing " of the known distances 
and angles involved in the problem, and measur- 
ing or " scaling " the unknown on the drawing. 

2. To do this we need to know how to measure 
angles as well as lines. Much practice has been 
given in the use of (1) the angular unit, i.e. the 
degree ; and (2) the instrument for measuring 
angles : the protractor. 



REVIEW EXERCISE 30 

1. A boy knows that AB is 100 ft. and that 
Z. B = 40°. From this information can he con- 
struct a scale drawing for the triangle ? Give 
reasons for your answer. 



lOOft. 




2. What facts or data must be known about a tri- 
angle before you can make a scale drawing of it ? 

3. A tree 90 ft. high casts a shadow 140 ft. long. 
Find from a scale drawing the angle of eleva- 
tion of the sun. 



Finding Unknown Distances by Scale Drawings 73 

4. Give the meaning of each of the following 
terms : exponent, factor, multiple, equation, 
scale, angle, and formula. 

5. Represent in the briefest way the product of 
five x's ; the sum of five xs. 

6. Does a 2 b = ab 2 if a is 4 and b is 3 ? 

7. Can you construct a scale drawing for the tri- 
angle in Fig. 40 if you know that AB is 200 ft., 
Z B = 40°, and A C = 150 ft. ? Why ? 

8. What must be known about a triangle before 
you can construct it accurately ? 

9. Find* in 6*- 2 = 19. 

10. Solve f or y : 16 + 5 y = 50. 

11. A man earned p dollars per month, and spent 
s dollars per week. Make a formula for his 
yearly savings. 

12. Evaluate the formula in example 11 when p = 
$150 and s = $25. 

13. Make a more complete summary of this chapter 
than is given above. 

14. If a square is made 6 inches wider, and 5 inches 
longer, the resulting figure will then have a 
perimeter of 122 inches. How much more area 
will it then have ? 

15. A train is traveling x miles per hour. If it in- 
creases its speed 5 miles per hour, it will then 
travel 240 miles in 6 hours. Find its original rate. 

16. One number is 8 more than another. Two 
thirds of the smaller, plus the larger, equals 28. 
Find each number. 



74 Fundamentals of High School Mathematics 



17. 



18. 



19. 



Angle A in a certain triangle is two fifths of 
angle B. Find each angle, if the third angle is 
54°. 

In an isosceles triangle, the base is two thirds of 
each of the equal sides ; its perimeter is 80. 
Find the base: 

A certain angle is bisected. One of the angles 
obtained lacks 30 degrees of being half of a 
right angle. How large was the angle which 
was bisected ? 



CHAPTER V 



A SECOND METHOD OF FINDING UNKNOWN 
DISTANCES: THE USE OF SIMILAR TRIANGLES 

Section 29. Scale drawings are somewhat inaccurate. 
Hence it is better to use similar triangles. In the last chapter 
we saw that scale drawings could be used to find unknown 
distances, either linear or angular. The results obtained, 
however, were very inaccurate. Seldom did many of you 
get the same answer for any example. Therefore we 
need more accurate methods for determining unknown lines 
and unknown angles. This chapter, and the next one, 
will show methods that depend less upon the accuracy or 
skill of the person who " scales " or measures. The first 
method, based upon geometrical figures of exactly the same 
shape, will be explained now. 

Section 30. What are similar figures? You have al- 
ready seen many objects or figures of exactly the same 
shape. A scale drawing has the same shape as the figure 
from which it was made ; on a photographic plate the figure 
is the same in outline or shape as the original ; the map of 
a state has the same shape or outline as the state itself. 

Figures which have the same shape are said to be similar 
figures. Which of the following figures are similar in 
shape ? 




Fig. 11 



Fig. 42 



Fig. 45 



Fig. 46 



75 



76 Fundamentals of High School Mathematics 

Similarity in shape in geometrical figures is a very im- 
portant principle that we are able to use in many ways in 
mathematics. Before this can be taken up, however, you 
need to know more about RATIO than you learned in arith- 
metic. This important method of comparing quantities 
will now be taken up. 



A NEW WAY TO COMPARE TWO QUANTITIES 
TO FIND THEIR RATIO 



NAMELY, 



Section 31. We have been comparing two quantities by 
finding how much larger or smaller one quantity was than 
another quantity. For example, if the line AB is 4 units 
long and the line CD is 6 units long, we should have said, 
in describing the comparative lengths, that the line AB 
was 2 units shorter than line CD, or that line CD was 
2 units longer than the line AB. 



Here the unit is \ inch. 



B 



D 



Fig. 47 

Another method, however, of comparing quantities is 
used very extensively in mathematics. It is the method of 
dividing one quantity by the other, or finding the quotient 
of the two quantities. Thus, to compare the line AB with 
the line CD we divide AB by CD, which gives : 

AB = 4t 
CD 6' 
This result is read "the quotient of AB and CD, or the ratio 
of AB to CD is equal to four sixths." This process is de- 
scribed as finding the ratio of the two lines. The ratio of 
two numbers means, then, the quotient which results from 
dividing one of the numbers by the other. Thus, the ratio 
of 5 to 10 is J, and is written -^ = ^. The ratio of 10 to 5 



Finding Unknown Distances by Similar Triangles 77 

is 2, and is written -5°-= 2. In the same way the ratio of 

1 in. to 1 ft. is -^2 ; the ratio of | to f is J; and the ratio 

4 x 4 
of ^x to 7 x is - 1 - or -• 
lx 7 

EXERCISE 31 
PRACTICE IN DEALING WITH RATIOS OF NUMBERS 

1. What is the ratio (in lowest terms) of 10 to 12 ? 
of 20 to 24 ? of 15 to 18 ? of 25 to 30 ? Show 
that 5 x and 6 x represent all pairs of numbers 
whose ratio is f . 

2. Give several pairs of numbers having the ratio \. 
Show that 3 x and 4 x represent all pairs of num- 
bers having this ratio. 

3. The ratio of two numbers, a and b, is \ . What 
is b when a is 40 ? 

4. The ratio of two lines, m and ;z, m 
is f . Find the length of m if n n 
is 18 in. 



Fig. 48 

5. Divide 40 into two parts whose ratio is |. 

6. A father and his son agreed to divide the profits 
from their garden in the ratio of \. Find each 
one's share if the total profits were 1210. 

7. The ratio of Z A to Z B is f . Find Z B when 
Z A is 80°. 




Fig. 49 



78 Fundamentals of High School Mathematics 



8. The ratio of the areas of the two squares, 5 2 and 



5 



is A 



Find the area of each if the sum of 



their areas is 45 sq. in. 



Si s. 



Fig. 50 



9. Divide an angle of 90° into two angles having 
the ratio of 4 to 5. 
10. Measure each angle, Fig. 51, with a protractor. 
Find their ratio. 




Fig. 51 

11. In the two triangles ABC and XYZ, what is the 
ratio of Z C to Z Z? of Z A to Z X? of Z ^ 





Fig. 52 



Finding Unknown Distances by Similar Triangles 79 

to Z F? Do these triangles have the same 
shape ? Do all triangles have the same shape ? 

12. Find the ratio of two lines if one is 2 feet long 
and the other 3 yards long. 

13. What is the ratio of 3 pints to 4 quarts ? 

14 ?-£■ 15. |=f- 16. I=™. 

n lo 3 6 7 « 

17. Find the ratio of AB to CD, Fig. 53, by measur- 
ing the length of each line. Express the result 
decimally. 

A B 



Fig. 53 

18. A school baseball team won 7 of the 10 games 

it played. What was its standing in percentage ? 

Solution : The standing of the team in percentage means 
the ratio of the number of games won to the number played, 
expressed decimally. Thus, the standing of this team is ex- 
pressed by the result obtained from changing the ratio, jfo, 
to a decimal. This gives .70 or .700. 

19. What was the percentage or standing of a team 
which won 9 of its 12 games ? 

20. The winning team in one league won 14 of its 
18 games ; and the winning team in another 
league won 15 of the 19 games it played. Which 
league had the higher percentage ? 

21. Express each of the following ratios as a deci- 
mal, correct to two places : 



^ I w I 


«*£ 


, , 21 
( ^28 


99 

U) 81 


<*>? <'4 


c/)| 


{k) 46 


, ., 16.2 



8o Fundamentals of High School Mathematics 



22. If 12 quarts of water are added to 25 gallons 
of alcohol, what is the ratio of the water to the 
entire mixture ? Express decimally. 
Section 32. Similar triangles. The next exercise is in- 
tended to show a very important fact about triangles which 
have the same shape, i.e. about similar triangles. 

EXERCISE 32 
PRACTICE WITH SIMILAR TRIANGLES 

1. Draw a line XY twice as long as *AB, Fig. 54. 
At X draw an angle equal to angle A. At F 
draw an angle equal 
to angle B. Produce 
the sides of these 
angles until they meet 
at Z. Measure the 
angle formed by these 
sides. How should it 
compare with angle C ? 
Why? 

2. \a) Angle A corre- 
sponds to what angle in your triangle ? 
(3) Angle B corresponds to what angle in your 
triangle ? 

(c) What is true, then, about the corresponding 
angles of the two triangles ? 

3. Measure the side in your triangle which corre- 
sponds to A C, and the side which corresponds to 
BC. What is the ratio of AB to XY, or what is 




the value of ? 



of 



AC 



? of 



BC 



? What does 



XY XZ YZ 

this tell about the ratios of corresponding sides ? 



Finding Unknown e Distances by Similar Triangles 8r 




4. Construct a tri- 
angle larger than 
Fig. 55 but having 
its angles equal to 
the angles of Fig. 
55. Is your tri- 
angle the same 
shape as Fig. 55 ? FlG - 55 

After careful measurement find the ratio of AB 
to its corresponding side in your triangle. Then 
find the ratio of AC to its corresponding side. 
Compare these ratios. 

IMPORTANT PRINCIPLE 

The previous experiments in finding the ratios of the 
sides of triangles of the same shape show the following 
important principle : 

If it is known that the angles of one triangle are 
equal respectively to the angles of another tri- 
angle, then it follows that the ratios of the cor- 
responding sides are equal. 

This principle or truth is used very much in mathe- 
matics. To illustrate, suppose we know that the angles of 
one triangle are equal respectively to the angles of another 
triangle ; then we also know that the ratios of correspond- 
ing sides are equal. Hence, we can make an equation from 
these equal ratios and from this equation find important 
unknown distances. The next exercise will show how this 
is applied to finding the length of lines. 

A proof, which is not based on measurement, is com- 
monly given for this fact, in the second year's work in. 
mathematics. 



82 Fundamentals of High School Mathematics 



EXERCISE 33 



ADDITIONAL PRACTICE WITH SIMILAR FIGURES 

1. Figure 56 is a 
right triangle. 
Why? If angle 
A is 30°, find an- 
gle C. 

2. In a right triangle 
one of the acute 
angles (that is, 
one of the angles smaller than a right angle) 
is 40°. Find the other acute angle. 



4. 




Fig. 56 






■* 








I 










w 










v 


90° 

\ 




37° 

* 




K 


«e ■ 


- 8 




— 1 



Fig. 57 



Fig. 58 



3. Figures 57 and 58 are right triangles. If angle 
A = angle D, are the triangles similar ? Why ? 
In Figs. 57 and 58, if EF=6, ED =8, 
and CB = 3, what must AB equal ? To solve 
this problem we use the principle that the ratios 
of the corresponding sides of similar triangles 

are equal. This gives the 

What, therefore, is AB ? 



_. x 3 

equation - = -. 



Finding Unknown Distances by Similar Triangles 83 

5. The sides of a triangular plat of ground are 
150 ft., 100 ft., and 125 ft., respectively. The 
side of a scale drawing of this plat, correspond- 
ing to the 150-foot side, is 5 cm. Find the side 
of the scale drawing corresponding to the 
100-foot side. Solve as in Example 4. 

6. The sides of a triangle are 3, 4, and 5 cm. The 
shortest side of a similar triangle is 16 cm. 
Find the other sides of the second triangle. 

7. A house is 36 ft. high and the garage is 16 ft. 
high. If the house is represented in a drawing 
as 18 in. high, how high should the drawing 
of the garage be ? What mathematical prin- 
ciple is used to show this ? 

8. Two rectangular gardens are the same shape, 
but of different size. The larger one is 72 ft. 
by 84 ft. If the length of the smaller one is 
40 ft., what must be its width ? 

9. Two angles of one triangle are equal respec- 
tively to two angles of another triangle. Are 
the triangles similar ? Why ? 

10. Line AB is parallel to line CD. Woufd they 
meet if produced, either to the right or to 
the left of 
the third line 
MN} Meas- 
ure Z. 1 and 
Z 2. These 
angles are 
called cor- 
responding Fig. 59 
angles of parallel lines. 




84 Fundamentals of High School Mathematics 



11. In Fig. 59 measure another pair of correspond- 
ing angles, Z 3 and Z 4. What do you find ? 

These two exercises illustrate a very important fact in 
mathematics ; namely, that the corresponding angles of 
parallel lines are always equal. Later on this will be 
proved without measuring the angles ; that is, without any 
possibility of error. You will make use of this fact with- 
out again measuring the angles. £ 

12. In triangle ABC, DE is drawn 
parallel to AB. DoesZl=Z2? V/ 1 
Why ? Is triangle DEC similar p^&- 
to triangle ABC? Why ? Fig. 60 

13. In Example 12, DC = 12, AC= 21, and CE = 14. 
Show how BC can be found, by using the prin- 
ciple that the ratios of corresponding sides of 
similar triangles are equal. What is the length 
of BC? 

14. A boy wishes to measure the height of a tree. 
He notes that the tree, AC, its shadow, AB, 

c 





Fig. 61 



and the sun's ray, CB, passing over the top of 
the tree, form a triangle. He measures the 
shadow and finds it 100 ft. long. At the same 



Finding Unknown Distances by Similar Triangles 85 

time a vertical stick 4 ft. high makes a shadow 
10 ft. long. Why is the triangle formed by the 
stick, its shadow, and the sun's ray passing over 
the top of the stick similar to the other triangle ? 
How can the boy find the height of the tree 
from the similar triangles ? What is its height ? 

15. A Boy Scout wagered he could find the distance 
between two trees, A and B, on opposite sides 
of a river, without crossing it. Could he do it, 
and if so, how ? If not, why not ? 

16. A crude way to measure the height of an ob- 
ject is by means of a mirror. Place a mirror 
horizontally on the ground at M, and stand at 
the point at which the image of the top of the 




Fig. 62 

object is just visible in the mirror. Show how, 
by measuring certain distances, this would 
enable one to compute the height of the object. 

17. In triangle ABC, DE is parallel to CB. Show 
that triangle A ED is similar to triangle ABC. 
If BC= 10, ED = 5, and AE = 8, what is AB ? 
Draw the figure. 

18. Figure 63 shows two triangles, with the size 
of each angle indicated, which a teacher drew 



86 Fundamentals of High School Mathematics 



19. 



upon the blackboard for an examination. She 
asked the following questions about the two 
triangles : 

(a) Are they similar triangles ? Why ? 

(6) Does ff=|r Wh y ? 




Fig. 63 



(c) Does — = 



BC 



(d) Does —tt> — 



Why? 
Why? 



ZY 

xz 7 

AB BC 

(e) Does the ratio of any two sides equal the 
ratio of any other two sides ? 
How would you have answered these questions ? 
The sides of a small triangle are 3, 4, and 6. 
Do you think it is similar to a larger triangle 
whose sides are 15, 18, and 30 ? 



SUMMARY OF THE IMPORTANT POINTS OF CHAPTER V 

It is important to have clearly in mind the following im- 
portant conclusions from the chapter : 

1. If you know that the angles of one triangle are 
equal respectively to the angles of another tri- 
angle, then you know that the ratios of the corre~ 



Finding Unknown Distances by Similar Triangles 87 

sponding sides are equal. In other words, you can 
make an equation, and thereby find an unknown 
side. 

2. The corresponding angles of parallel lines are 
equal. 

3. Unknown distances may be found by means of 
similar triangles. 

REVIEW EXERCISE 34 

1. Translate into words : 4-y -f 3 =y + 21. 

2. If A, B, and C represent the number of degrees 
in the respective angles of a triangle, we know 
that A + B + C= 180°. Why ? What is A if 
^ = 40° and C= Q5°? 

3. If five times a certain number is divided by 2.7, 
the result is 3. What is the number ? 

4. Given the formula V— Iwh, find a formula for 
/; for w. 

5. A boy receives C cents an hour for regular work, 
and pay for time and a half when he works over- 
time. What will represent his earnings for 
6 hr. overtime? Evaluate this when £7=50. 

6. If n represents a boy's present age, what is 
the meaning of the expression n — 5 = 11 ? of 
n + 4 = 20 ? 

7. The sides of one triangle are 10, 12, and 15 
inches ; the sides of a similar triangle are 8, x, 
and 12 inches. Determine the length of x, the 
side which corresponds to the 12-inch side in 
the first triangle. 



88 Fundamentals of High School Mathematics 



9. 



10. 



11. 



12. 



One number is three times as large as another ; 

two fifths of the large number added to one 

half of the smaller number gives 17. Find 

each number. 

The second angle of a triangle is 15° less than 

the first angle ; the third angle is one sixth of 

the first. Find each angle. 

One number is 5 larger than another; their 

ratio is f . Find each number. 

Room A in a certain school gave twice as much 

to the Red Cross as Room B ; Room C gave 

1 4 more than half as much as Room B. What 

was given by each room, if all three gave $ 39 ? 

Does x 2 + 5 x = 14 make an equation if x = 2 ? 

if*=3? if jr = 4 ? 



CHAPTER VI 

HOW TO FIND UNKNOWNS BY MEANS OF THE RIGHT 

TRIANGLE 

Section 33. The advantage of the RIGHT TRIANGLE in 
finding unknowns. It should be clear by this time that 
mathematics gives us methods of finding unknown quanti- 
ties. The equation is the most important tool for doing 
this, for the reason that when we solve a problem we have 
to make an equation. This equation must contain the un- 
known quantity together with other known quantities 
which are related to it in some way. 

In the last chapter we saw that an equation could be 
formed from the ratios of corresponding sides of similar 
triangles and that by that means we could find an un- 
known length. Two facts, however, make that method less 
satisfactory than the one we shall study in this chapter : 
(1) we must always be certain the triangles are similar, or 
we have no right to make an equation, and (2) the method 
is cumbersome because we must always use two triangles. 

There is a particular kind of triangle whose properties can 
be used to find unknown distances accurately and at the same 
time more easily than by any other method. It is the right 
triangle. The most important facts about the right triangle 
are found in connection with first, the ratios of its different 
sides ; second, the relation between the hypotenuse and the 
other two sides. We shall first discuss the ratios of the 
different sides. 

A. FINDING UNKNOWNS BY USING THE RATIOS OF 
THE SIDES 

I. THE TANGENT OF AN ANGLE 

Section 34. The ratio of the " side opposite " a given 
angle to the " side adjacent " the given angle, i.e. the 

8 9 



90 Fundamentals of High School Mathematics 

TANGENT of the angle. You will recall that in a right tri- 
angle one angle is 90° and the sum of the two acute angles 
equals 90°. (Why ?) 
In finding unknown 
distances by means of 
right triangles we shall 
always deal especially 
with one of the acute 
angles. Therefore, in 
referring to the sides of 
a right triangle, when 
dealing with a given 




side adjacent to 30 °L 

Fig. 64 



angle, we shall speak of them as they are described in 
Figs. 64 and 65. If angle B is the acute angle with which 




hypotenuse 



Fig. 65 



we are concerned, then side AC is the " side opposite" 
Z B, and side AB is the "side adjacent" Z B. The side 
opposite the 90° angle is always called the hypotenuse. 
Some exercises will show the importance of the ratio of 

the "side opposite " 
the "side adjacent" 

an acute angle of a right triangle. 



Finding Unknowns by Means of Right Triangle 91 



EXERCISE 35 

SOME MEASUREMENTS TO FIND THE NUMERICAL VALUE OF THE RATIO 
OF THE ■ ' SIDE OPPOSITE " ' TO THE « ' SIDE ADJACENT " A 30° ANGLE 
OF A RIGHT TRIANGLE 



1. 




Fig. 66 

In Fig. 66, a is the " side opposite " the 30° angle 
and b is the "side adjacent" the 30° angle. 
Measure a and b. Now find the numerical value 
of the ratio of a to b by dividing the length of 
a by the length of b. Record your results in 
Table 1. 

In Fig. 67, a and b are respectively the " side 
opposite" and the "side adjacent" an acute 
angle of 30°. Measure each and compute the 

ratio -j to two decimal places. Record your 
results in Table 1. 




92 Fundamentals of High School Mathematics 

3. Draw any other triangle similar to those above, 
but with much larger sides. Measure the " side 
opposite " and the " side adjacent" the 30° angle 
and compute their ratio as before. Record re- 
sults, as before, in Table 1. 

Table 1. Record here the results of measuring 
the sides of right triangles and of computing the 
ratio of the "side opposite" to the "side adja- 
cent " an acute angle of 30°. 

Table 1 





Length of a 


Length of b 


Ratio of a tob (5.e.,ir) 


Fig. 








% 








Fig. 








Fig. 









4. 



What do you notice in the table about the nu- 
merical values of the ratios 

the "side opposite" an acute angle of 30° a 

the " side adjacent" an acute angle of 30° b' 

The members of the class should compare re- 
sults, to see what result seems most likely to be 
the true one. If great care is taken in measur- 
ing, the ratio should be very close to .58 in each 
triangle. Why should it be the same in each 
triangle ? 

In Fig. 68, CB, DE, and GF are perpendicular to 
AB. Is triangle AFG similar to triangle AED? 
Why ? Is either of the smaller triangles sim- 



Finding Unknowns by Means of Right Triangle 93 




ilar to the large triangle ? Why ? From this, why 

C F 
does the ratio of GF to AF, or -— - , equal the 

A r 

DE 

ratio of DE to AE, or — — ? If you measured 

Ah J 

these lines, and computed the ratios, what would 

you expect to be true of the results ? 

Section 35. This last example is very important, because 

GF 

it shows, without measurement, that the ratio equals 



the ratio 



DE 
AE 



But this is the same as saying that the 



ratio of the "side opposite" to the "side adjacent" a 30° 
angle in one right triangle is ALWAYS equal to the ratio of 
the "side opposite" to the "side adjacent" a 30° angle in 
any other right triangle. The length of the sides may be 
far different, but the ratios should be the same. This shows 
that the ratios obtained in the table should have been the 
same, if it were possible to draw and measure without error. 
We shall now make use of the fact that the numerical 
value of the ratio of the "side opposite" to the "side ad- 
jacent" an acute angle of 30° (in a right triangle) is ap- 
proximately .58. 



94 Fundamentals of High School Mathematics 



EXERCISE 36 

1. Illustrative example. 

A man wishes to determine the height of a smokestack. 
He finds that the angle of elevation of the top of the 
smokestack, from a point 200 ft. from the base of the 
smokestack, is 30°. 




Fig. 69 



Solution 



h 

200 

h 

h 



= .58. (Why?) 

= 200x.58. (Why?) 
= 116 ft. 



Note here that is the ratio of the " side opposite " to 

200 ™ 

the "side adjacent " the 30° angle. From previous work 
we know that this ratio is .58. Thus, we can make the 

equation -^-=.58. 
H 200 

2. In triangle ABC, angle A is 30° and angle C is 
60°. Find CB if AB is 70 yards. What is CB 
if AB is 10 inches ? Draw the figure. 

3. In triangle XYZ, angle X is 30° and angle Z 
is 60°. Find XY if YZ is 116 ft. 

4. In triangle DEF, angle E is 30° and angle D 
is 90°. Find ED if DF is 24 in. 



Finding Unknowns by Means of Right Triangle 95 

5. In Fig. TO, CD bisects angle C and is perpen- 
dicular to AB. How many degrees in angle 
BCD ? If CD is 100 cm., how long is DB ? 




6. In the right triangle ABC, angle B is 60°, angle 
A is 30°, and BC is 50 ft. Find AC 

7. In triangle XYZ, angle X is 30°. Does the 

ZY 

ratio = .58 ? State definitely when this 

JC Y 

ratio is equal to .58. 




Fig. 71 

Section 36. It is convenient to name important ratios. 

Since it is helpful to use the ratios of the various sides of a 
right triangle, very frequently in finding unknown dis- 
tances, each is given a definite name. The ratio of the 
"side opposite" an acute angle to the "side adjacent" is 
called : 



96 Fundamentals of High School Mathematics 



THE TANGENT OF THE ANGLE 

Its abbreviation is tan. Thus, in the above examples 
the tangent of an angle of 30° is constant; it is approxi- 
mately .58. 

EXERCISE 37 

1. Construct a right triangle such as Fig. 72, with 
AB equal to 4 cm. and angle A equal to 40°. 
Then measure BC and from that Q 
find the tangent of an angle of 
40°. Compare results with those 
of other members of the class. 

2. In a similar way 
find the tangent 
of an angle of 
50°. (Use AB A 
as 4 cm.) Also 
find the tangent 
of each of the fol- 
lowing angles : 60°, 70°, and 20°. 

Section 37. Summary of steps in finding the tangent of 
an angle. These examples show how to find the tangent of 
any angle. Three steps are necessary ; namely : (1) meas- 
ure the side opposite the particular angle ; (2) measure the 
side adjacent the angle ; (3) divide the first number ob- 
tained by the second. To do this, however, for angles of 
all sizes from very small to very large, would require a 
great deal of labor, and probably give, for a great many of 
you, inaccurate results. To save this trouble, and at the 
same time get very accurate results, these ratios or tangents 
have been computed very carefully and compiled in a table 
like Table 4. (See Table of Tangents on page 97.) 




Finding Unknowns by Means of Right Triangle 97 

Table 2 
TABLE OF COSINES AND TANGENTS 

Numerical Values of the Tangents and Cosines of the Angles 
from 0° to 90° Inclusive 



Deg. 


tan 


cos 


Deg. 


tan 


cos 





.000 


1.000 


46 


1.04 


.695 


1 


.017 


.999 


47 


1.07 


.682 


2 


.035 


.999 


48 


1.11 


.669 


3 


.052 


.999 


49 


1.15 


.656 


4 


.070 


.998 


50 


1.19 


.643 


5 


.087 


.996 














51 


1.23 


.629 


6 


.105 


.995 


52 


1.28 


.616 


7 


.123 


.993 


53 


1.33 


.602 


8 


.141 


.990 


54 


1.38 


.588 


9 


.158 


.988 


55 


1.43 


.574 


10 


.176 


.985 














56 


1.48 


.559 


11 


.194 


.982 


57 


1.54 


.545 


12 


.213 


.978 


58 


1.60 


.530 


13 


.231 


.974 


59 


1.66 


.515 


14 


.249 


.970 


60 


1.73 


.500 


15 


.268 


.966 














61 


1.80 


.485 


16 


.287 


.961 


62 


1.88 


.469 


17 


.306 


.956 


63 


1.96 


.454 


18 


.325 


.951 


64 


2.05 


.438 


19 


.344 


.946 


65 


2.14 


.423 


20 


.364 


.940 














66 


2.25 


.407 


21 


.384 


.934 


67 


2.36 


.391 


22 


.404 


.927 


68 


2.48 


.375 


23 


.424 


.921 


69 


2.61 


.358 


24 


.445 


.914 


70 


2.75 


.342 


25 


.466 


.906 














71 


2.90 


.326 


26 


.488 


.899 


72 - 


3.08 


.309 


27 


.510 


.891 


73 


3.27 


.292 


28 


.532 


.883 


74 


3.49 


.276 


29 


.554 


.875 


75 


3.73 


.259 


30 


.577 


.866 














76 


4.01 


.242 


31 


.601 


.857 


77 


4.33 


.225 


32 


.625 


.848 


78 


4.70 


.208 


33 


.649 


.839 


79 


5.14 


.191 


34 


.675 


.829 


80 


5.67 


.174 i 


35 


.700 


.819 














81 


6.31 


.156 


36 


.727 


.809 


82 


7.12 


.139 


37 


.754 


.799 


83 


8.14 


.122 


38 


.781 


.788 


84 


9.51 


.105 


39 


.810 


.777 


85 


11.4 


.087 


40 


.839 


.766 














86 


14.3 


.070 


41 


.869 


.755 


87 


19.1 


.052 


42 


.900 


.743 


88 


28.6 


.035 


43 


.933 


.731 


89 


57.3 


.017 


44 


.966 


.719 


90 


Inf. 


.000 


45 


1.000 


.707 









98 Fundamentals of High School Mathematics 

EXERCISE 38 
FINDING ANGLES AND TANGENTS FROM THE TABLE OF TANGENTS 

Find, from Table 4, each of the following : 

1. tan 42°. 

2. The angle whose tangent is .58. 

3. tan 57°. 

4. The angle whose tangent is .94. 

5. tan 14°, 

6. The angle whose tangent is -|. 

7. tan 25°. 

8. The angle whose tangent is f. 

9. tan 45°. 

EXERCISE 39 



EXAMPLES WHICH INVOLVE THE USE OF THE TANGENT OF AN ANGLE 

1. Illustrative example. The brace wire AC of a telephone 
pole BC, Fig. 73, makes with 
the ground an angle of 62°. 
It enters the ground 15 ft. 
from the foot of the pole. 
Find the height of the pole BC. 
Solution : 

— = tangent 62°. 
AB 



BC 
15 



= 1.88 (from the Table). 



2. 



BC = 15 x 1.88 = 28.2. 

The angle of elevation 
of the top of a tree, from 
a point 75 ft. from its 
base (on level ground), 
is 48°. How high is the 
tree ? 




Fig. 73 



Finding Unknowns by Means of Right Triangle 99 



3. From a vertical height of 1500 yd. a balloonist 
notes that the angle of depression of the enemy 
trench is 51°. Find the distance from the trench 
to the point on the level ground directly below 
the balloonist. Make a drawing. 

4. The angle of elevation of an aeroplane at point A 
on level ground is 44°. The point B on the 
ground directly beneath the aeroplane is 450 yd. 
from A. How high is the aeroplane? 

5. If a flagpole 42 ft. high casts a shadow 63 ft. 
long, what is the angle of elevation of the sun ? 

6. In Fig. 74, CD is per- 
pendicular to AB. 
Find AD if angle 
^=60° and CD = 20. 

7. From the point of 
observation on a mer- 
chant vessel, the an- 
gle of depression of 
the periscope of a 

submarine was 17°. How far was the submarine 
from the merchant vessel, if the observer was 
40 ft. above the water ? 

8. Turn back to page 71 and solve problem 13 by 
this method. How do your results compare with 
those obtained by scale drawings ? 

9. Solve Example 12, page 70, by this method. 
Which method is preferable ? 

Section 38. There are many problems which cannot be 
solved by means of the tangent. In the previous section 
we found that the ratio of the " side opposite " to the 




ioo Fundamentals of High School Mathematics 



"side adjacent" an acute angle of a right triangle is 
always constant for any particular angle. This enabled 
us to find the length of the sides and the size of the 
acute angle. Now we come to another fact about right 
triangles. Let us examine a problem which cannot be 
solved by the use of the tangent. 

Illustrative prob- 
lem. In Fig. 75, 
BC represents a tele- 
phone pole, AC an 
anchor wire, and AB 
the distance from the 
foot of the pole to 
the point at which 
the wire enters the 
ground, 20 ft. The 
wire makes an angle 
of 30° with the 
ground. How long 
is the wire ? 




II. THE COSINE OF AN ANGLE 

Section 39. The ratio of the "side adjacent" the given 
angle to the hypotenuse of the triangle, i.e. the COSINE. 

Clearly, AC, in the above example, cannot be found by 
means of the ratio which we called the tangent, because 
the tangent of 30° makes use only of BC and AB, and 
we must get a ratio which contains AC. Therefore, to 
solve this problem we shall have to learn how to use the 
ratio of the " side adjacent" the 30° angle, to the hypotenuse, 

AB 
0t AC- 



Finding Unknowns by Means of Right Triangle 101 



EXERCISE 40 
EXPERIMENTS TO DETERMINE THE NUMERICAL VALUE OF THE RATIO 

THE "SIDE ADJACENT " A 30° ANGLE JE TRE C OSINB 
THE HYPOTENUSE OF THE TRIANGLE' 




1. Measure the length of b and 



c in 



Fig. 76. 



Then compute the ratio - to two decimals. 

c 

Draw any other triangle similar to Fig. 76, but 
with much longer sides. Find, as in Example 1, 
the ratio of the side adjacent the 30° angle, 
to the hypotenuse. Compare your result with 
that of Example 1. 




3. In Fig. 77, EF and GH are perpendicular to AB. 



Why does 



AH 

AG' 



AF 
AE 



AB 
AC 



102 Fundamentals of High School Mathematics 

Section 40. The COSINE of a particular angle is CON- 
STANT. This -shows that the ratio of the " side adjacent " 
a 30° angle to the hypotenuse of one right triangle is 
equal to the same ratio in any other right triangle which 
has an acute angle of 30°. For this reason, you would get 

the same numerical value for - in Examples 1 and 2, if it 

c 

were not for errors in measurement. 

Therefore, just as in the case of the tangent, so the 

"side adjacent " 30° angle . , 

cosine, i.e. 1 ?— , is always constant, 

hypotenuse 

when the angle is 30°. It is approximately .86. The 

right triangles may differ in size and position, but as long 

as they are similar (that is, so long as the acute angles we 

are dealing with are the same size), this ratio does not 

change. 

EXERCISE 41 

PROBLEMS SOLVED BY APPLYING THE CONCLUSION ARRIVED AT ABOVE; 
NAMELY, THE RATIO OF THE " SIDE ADJACENT " A 30° ANGLE TO 
THE HYPOTENUSE IS .86. 

1. Illustrative example. The anchor wire AC, of a 
telephone pole, meets the ground 20 ft. from the foot of 
the pole, making an angle of 30° with the ground. Find 
the length of the wire AC. 



Solution : 


ff=.«8. (Why?) 






20 - SB 

T _.86. 

20 = . 86 h, or ft = 23.2 ft. 


y^ 




A^ 


30° 

, \ 



B 



Fig. 78 



Finding Unknowns by Means of Right Triangle 103 

2. The rope, AC, of the flagpole, BC, makes -an 
angle of 30° with the ground, at a point 42 ft. 
from the foot of the pole. How long is the 



3. 



5. 



rope ? Make a drawing. 

The angle of elevation of the top of a tree from 

a point A, on level ground, 100 ft. from the 

base of the tree, is 30°. What is the distance 

from A to the top of the tree ? 

In the right triangle ABC, AB is 25 in. and 

Z^ = 30°. Find A C 




Fig. 79 



Draw a right triangle such that angle A = 60° 
and the hypotenuse AC = 4 in. From this could 
you find BC1 
6. The angle of depression of a boat, from the top 
of a cliff, is 30°. Find the distance from the 
observer to the boat, if the boat is 400 ft. from 
the foot of the cliff. 
These examples have been solved by using the ratio 
of the "side adjacent" an acute angle of 30° to the 
hypotenuse, or, as we shall call it from now on, by using the 
cosine of the angle. The abbreviation for cosine is cos. Thus, 

side adjacent "Z^ 



cos Z. B = ratio of 



hypotenuse of the triangle 



io4 Fundamentals of High School Mathematics 



1. 



EXERCISE 42 

Construct a right triangle similar to Fig. 80, with 
A = 40° and AB = 4 cm. Then measure c and 

compute the ratio -• By comparing your 
c 

result with cos 40° as given in the table, see 
if you are within .05 of the correct result. 




k— "b = 4cm. 

Fig. 80 



B 



2. How would you construct or draw the cos of 
a 60° angle ? of an 80° angle ? 

3. Read from the table of cosines : 
O) cos 67°. 

(b) The angle whose cos is .258. 

\c) cos 45°. 

id) The angle whose cos is .573. 

0) cos 2°. ■ 

(/) The angle whose cos is .707. 

\g) cos 89°. 

(Ji) The angle whose cos is .629. 

4. A surveyor desires to measure the distance BC 
across a swamp. He surveys the line BA per- 
pendicular to BC. He extends this line BA until 
he can measure from A to C. If AC is 400 ft. 



Finding Unknowns by Means of Right Triangle 105 



6. 



7. 



and angle C is 55°, show how he would com- 
pute the length of BC. Find BC. 




Fig. 81 

A boy observes that his kite has taken all 
the string, 750 ft. Assuming that the string is 
straight and that it makes an angle of 34° with 
the ground, how far on level ground is it from 
the boy to the point directly below the kite ? 
The angle of elevation of the top of a tent pole, 
from a point 43.2 ft. from the foot of the pole, 
is 32°. Find the distance from the point of ob- 
servation to the top of the pole. 
Figure 82 is a right triangle. Find AB if 
angle A is 42° and AC = 67. Hint: What 
is the cosine of angle A ? 

C 




Fig. 82 



io6 Fundamentals of High School Mathematics 



9. 



10. 



11. 



12. 



How long a rope will be required to reach from 
the top of a flagpole to a point 19 ft. from the 
foot of the pole (on level ground) if the rope 
makes an angle of 63° with the ground ? 
The angle of depression of a boat from the top 
of a cliff is 37° when the boat is 1260 ft. from 
the foot of the cliff. Find the distance from 
the boat to the top of the cliff. 
Find angle A if AB is 27 and AC is 48. 
Hint : What is \\ with respect to angle A ? 




Fig. 83 

A man starts at O and travels in a direction 
which is 48° east of a north-south line. How 
far due north of O will he be when he is 26 
miles from O ? 

From the table find the cosine of 32°. Then 
find the cosine of an angle twice as large as 32°, 
and see if it is twice as large as the cosine of 
32°. Does the cosine of an angle change or 
vary in the same way that the angle changes 
or varies ? 



Finding Unknowns by Means of Right Triangle 107 



EXAMPLES IN WHICH NEITHER THE COSINE NOR THE TANGENT OF THE 
GIVEN ANGLE MAKES USE OF THE UNKNOWN SIDE 

The cosine of a given angle may not make use of the side 
which is unknown. It is then necessary to use the other 
acute angle. 

Illustrative example. To what height on a vertical wall 
will a 38-foot ladder reach, if it makes an angle of 58° with the 
ground? ^ 

In an example like this one, we 
cannot use either the cosine of 58° 
or the tangent of 58°, because 
neither makes use of the two 
lines, BC and AC. It is neces- 
sary to use the other acute angles 
A C, which in this case is 32°. 
(Why?) Then we can use the 

cosine of ZC, i.e. — . Thus we 
38 

have the equation 
x 



38 



= .848. 




Fig. 84 



EXERCISE 43 



Find ZY in right tri- 
angle XYZ, Fig. 85. 



2. Determine the length 
of BC in right tri- 
angle ABC, Fig. 86. 




Fig. 85 



108 Fundamentals of High School Mathematics 



3. 



4. 



A man travels from O in a direction which is 
50° east of a north-south line. How far is he 
from the north-south line when he has traveled 
60 miles from the starting point, O ? 
An aviator, 4200 yd. directly above his own 
lines, takes the angle of depression of the 
enemy's battery. What must be the range of 
the enemy's machine guns to endanger him, if 
the angle of depression is 29° ? 



B. FINDING UNKNOWNS BY THE RELATION BETWEEN 
THE HYPOTENUSE AND THE OTHER TWO SIDES 
OF A RIGHT TRIANGLE; THAT IS, BY 

THE HYPOTENUSE RULE 

Section 41. Previous use of the right triangle. We 

have already seen how right triangles can be used to find 
unknown distances. If we knew one side, and one acute 
angle, we were able to find any other side. Now we come 
to another method of dealing with right triangles ; namely, 
when two sides are known, but when no acute angle is known. 
This method will be illustrated by the following problem : 

Illustrative problem. 
What is the longest straight 
line you can draw upon a rec- 
tangular blackboard 28 in. 
wide and 36 in. long ? 

Evidently the longest straight 
line is the diagonal of the 
blackboard, or the hypotenuse 

OF THE RIGHT TRIANGLE, Fig. 

87. Thus, we need to know 
how to find the hypotenuse of 
a right triangle when the other 
two sides are known. This leads to the following i 




Finding Unknowns by Means of Right Triangle 109 



HYPOTENUSE RULE, OR LAW 

which states that the square of the hypotenuse of a right 
triangle is equal to the sum of the squares of the other two 
sides, or, in the above problem, that 
h 2 = 36 2 + 28 2 . 

This relation between the sides of a right triangle can be 
seen from Fig. 88. The 
base, AB, and altitude, AC, 
of a right triangle are 
drawn so that they contain 
a common unit an integral 
number of times. AB con- 
tains the common unit 4 
times and AC contains it 
3 times. Then by actual 
measurement BC will con- 
tain the same unit 5 times. 
By constructing squares on 
the sides of the triangle, 
you can see by counting that 
the sum of the squares on AB and AC is equal to the 
square on BC. 

To test this further, the pupil should construct a 
right triangle with the base 12 units and the altitude 
5 units. Then actually measure the hypotenuse, and 
note whether the square on the hypotenuse is equal 
to the sum of the squares of the other two sides. Now 
we are ready to go back to the problem of finding the 
longest line that can be drawn upon the blackboard. 
By making use of the truth which was just studied we 
have : 



























A 









































Fig. 88 



no Fundamentals of High School Mathematics 

h 2 = 28 2 + 36 2 
or h 2 = 784 + 1296 
or h 2 = 2080 



or /;=V2080 = 45.66 in. 

This relation between the sides of a right triangle is 
more widely used by engineers, carpenters, mechanics, and 
builders than any other mathematical law. Historical 
records show that a knowledge of this important relation 
is nearly as old as civilization itself. It is often called the 
Pythagorean Theorem, because of the fact that Pythagoras, 
a celebrated Greek mathematician, was the first to give a 
real proof for it. 

Note that it was necessary to be able to find the square 
root of a number in order to find the hypotenuse. That 
is, we had to solve the equation 

h 2 = 2080. 

To solve equations such as this, we must find the square 
root of each side. A brief review of square root will be 
helpful at this time. 

Section 42. Square root, as you did it in arithmetic. 
To enable you to use the hypotenuse rule well, you must 
be skillful in finding the square root of numbers. The 
illustrative example which follows will review the way you 
found square root in arithmetic. 

Illustrative example. Find the square root of 200. 

Note the following steps : 

(1) The number is separated into periods of two figures each, 
counting from the decimal point. 

(2) You find the greatest square in the left-hand period, and 
write its square root for the first figure of the root. 

(3) Subtract this square from the left-hand period, and with the 



Finding Unknowns by Means of Right Triangle in 



remainder place the next period for a 
new dividend. (This is 100 in the ex- 
ample.) 

(4) Double the part of the root already 
found (2 x 1 = 2) for your trial divisor. 
Divide the dividend, exclusive of the right- 
hand figure (10) by the trial divisor, 2. 
Write the quotient obtained, 4, as the next 
figure of the root and the divisor. Mul- 
tiply the complete divisor, 24, by the last 
term of the root, 4. Subtract the product, 
96, from the dividend, 100. To the re- 
mainder, 4, annex the next period, 00, for a new dividend. Re- 
peat this process until all periods are used, or until any required 
degree of accuracy is obtained. 



200.0000)14.14 
1 
24)100 . 
96 
281)400 
281 
11900 
2824 )11296 
""604 







EXERCISE 44 






PRACTICE 


IN FINDING SQUARE ROOTS 


OF NUMBERS 


1. 


1024 


8. 15625 


15. 11.6964 


2. 


1296 


9. 17161 


16. 372.49 


3. 


2025 


10. 19600 


17. 10 


4. 


3844 


11. 40401 


18. 50 


5. 


9801 


12. 47961 


19. 3 


6. 


5184 


13. 50625 


20. 2 


7. 


6241 


14. 358801 


21. 8 



EXERCISE 45 
PROBLEMS- BASED ON THE HYPOTENUSE LAW 

1. A rectangular schoolroom floor is 32 feet long 
and 28 feet wide. What is the longest straight 
line that could be drawn upon the floor ? 

2. How much walking is saved by cutting diago- 
nally across a rectangular plot of ground which 
is 25 rods wide and 42 rods long ? 



ii2 Fundamentals of High School Mathematics 

3. A tree 100 feet high was broken off by a storm. 
The top struck the ground 40 feet from the foot 
of the tree, the broken end remaining on the 
stump. Find the height of the part standing, as- 
suming the ground to be level. Make a drawing. 

4. What is the diagonal of a square whose sides 
are each 10 in. ? 

5. Find the side of a square whose diagonal is 20 
inches. 

6. Two vessels start from the same place, one sail- 
ing due northwest at the rate of 12 miles per 
hour, and the other sailing due southwest at 
the rate of 16 miles per hour. How far apart 
are they at the end of 3 hours ? 

7. The foot of a 36-foot ladder is 13 ft. 6 in. from the 
wall of a building against which the top is lean- 
ing. How high on the wall does the top reach ? 

8. A rope stretched from the top of a 62-foot pole 
just reaches the ground 16 feet from the foot 
of the pole. Find the length of the rope. 

SUMMARY OF IMPORTANT POINTS OF CHAPTER VI 

1. To find unknown distances by means of scale 
drawings is somewhat inaccurate and laborious; 
to use similar triangles involves two similar 
triangles, and the method is cumbersome. 

2. Hence we see the great advantage of the right 
triangle in finding unknown distances. It can 
be easily used in two ways : 

3. The first method is to find the ratio of one side 
of the triangle to another side. Two different 
ratios are used : first, the ratio of the " side 



Finding Unknowns by Means of Right Triangle 113 

opposite " a given angle to the " side adjacent," 
the tangent; second, the ratio of the " side 
adjacent " the given angle to the hypotenuse of 
the triangle, the cosine. 

4. The tangent and cosine of given angles are con- 
stant. 

5. Hence it has been found convenient to compute 
the tangents and cosines for all angles from 0° to 
90° and compile them in a table ; knowing the 
angle, therefore, one can read the tangent or 
cosine at once from the table. 

6. The second method of using the right triangle to 
find unknown distances is to use the hypotenuse 
rule for the relation between the hypotenuse and 
the other two sides of a triangle, namely : the 
square of the hypotenuse of a right triangle is equal 
to the sum of the squares of the other two sides. 

REVIEW EXERCISE 46 

In this list of problems you will have to decide for your- 
self whether to use the tangent or the cosine. Make a 
drawing for each problem ; indicate the parts that you 
know, and the part you are to find. 

1. A flagpole 50 ft. high casts a shadow 80 ft. 
long. What is the angle of elevation of the 
sun ? What time of year is it ? 

2. A searchlight on the top of a building is 180 ft. 
above the street level. Through how many 
degrees from the horizontal must its beam of 
light be depressed so that it may fall directly on 
an object 400 ft. down the street from the base 
of the building ? 



ii4 Fundamentals of High School Mathematics 

3. From the top of a cliff 120 ft. above the surface of 
the water, the angle of depression of a boat is 20°. 
How far is it from the top of the cliff to the boat ? 

4. At a time when the sun was 55° above the hori- 
zon, the shadow of a certain building was found 
to be 98 ft. long. How high is the building? 

5. A 40-foot ladder resting against a building 
makes an angle of 53° with the ground. Find 
the distance from the foot of the ladder to the 
building, and the distance from the top of the 
ladder to the base of the building. 

6. A man starts at O and travels in a direction 
which is 24° west of a north-south line through 
O, at the rate of 80 miles per day. At the end 
of 4 days how far north is he from an east-west 
line through O? How far west is he from a 
north-south line through O ? 

7. What direction will a boy be from his starting 
point if he goes 40 miles due north and then 
18 miles due east ? 

8. The gradient or slope of the railroad which runs 
up Pike's Peak is, in some places, 18%, z>.,in 
going 100 ft. horizontally it rises 18 ft. What 
angle does the road make with the horizontal? 

9. A searchlight on the top of a building is 180 
ft. above the street level. Through how many 
degrees from the horizontal must its beam of 
light be depressed so that it may shine directly 
on an object 800 ft. down the street from the base 
of the building ? How does your result compare 
with that obtained in Ex. 2, page 113 ? 



Finding Unknowns by Means of Right Triangle 115 



10. 



12. 



13. 



14. 



From the tenth story of a building the angle 
of depression of an object on the street level is 
30° ; from the eighteenth story the angle of de- 
pression of the same object is 50°. Find the 
distance from the object to the base of the build- 
ing if the second observation point is 80 ft. above 
the first observation point. Consider the eigh- 
teenth story 180 feet above the level of the street. 




Fig. 89 



11. Find AB and BC in Fig. 89, if AC = 100 ft. 



How would you pro- 
ceed to find the area 
of the triangle repre- 
sented in Fig. 90 ? 




Fig. 90 



Draw a triangle in which AB shall represent 
20 ft. and BC shall represent 18 ft. Is it 
correct to say that tangent A = -J -J ? Why ? 
An aviator observes that the angle of depres- 
sion of a machine gun nest is 50°. How far is 
it from a point on the level ground directly be- 
low the aviator, if he is 4000 feet high ? How 
far is the aviator from the nest ? 



CHAPTER VII 

HOW TO REPRESENT AND COMPARE QUANTITIES BY 
MEANS OF STATISTICAL TABLES AND GRAPHS 

Section 43. What we have already learned : How to 
solve equations and to find unknowns. We have now 
studied how to find unknown values by several methods. 
First, we have learned how to use the simple equation to 
find unknown numbers and how to construct practical 
formulas to aid us (Chapters I, II, and III). Then we 
learned how to find unknown distances by three methods : 
first, by making scale drawings (Chapter IV) ; second, 
by using the corresponding sides of similar triangles 
(Chapter V) ; and third, by using the sides of the right 
triangle (Chapter VI). In each of these latter methods 
we have also used equations. Thus you can already see 
the importance of being able to use the equation skill- 
fully. 

Section 44. Need for methods of comparing quantities. 
The chief aim of mathematics, however, is to help us in 
comparing quantities and in seeing how quantities are 
related to each other. For example, in conducting school 
work, we need to be able to compare the work of one 
school with another, or of one class with another. We 
need to be able to represent such facts as the progress of 
pupils in their studies, to compare the attendance in 
schools, or the population in cities, etc. 

Section 45. Facts of similar kind are called STATISTICS. 
When we have a great many quantities, like these, all of 
the same general sort, we speak of them as statistics, or 
statistical facts. To illustrate : Suppose you read in the 
paper that the size of the graduating class in your school 
last June was 97, and that for each ten years previous the 
number of graduates had been, 1909, 52 ; 1910, 57 ; 1911, 

n6 



How to Represent and Compare Quantities 117 

55; 1912, 63; 1913, TO; 1914, 75; 1915, 81; 1916, 83; 
1917, 91 ; 1918, 97. These figures are called statistical 
facts, or just "statistics:" 

Many illustrations can be found in everyday life of 
practical statistics : The school marks of children in a 
class ; the number of inhabitants in a list of cities, or in a 
given city for a number of years ; the heights of boys of 
a certain age ; the wealth of countries, or cities, or indi- 
vidual persons ; etc. Since we deal with such statistics so 
commonly it is helpful to have economical methods of 
expressing and comparing them. In this chapter we shall 
study these methods. 

Section 46. THREE IMPORTANT METHODS OF DEAL- 
ING WITH QUANTITIES IN MATHEMATICS 

There are three methods of representing facts, of compar- 
ing them, and of expressing the relationship between them. 

(1) The tabular method : in which we compile the 
facts in a table. 

(2) The graphic method : in which we make a 
graphic picture of the data. 

(3) The equational or formula method : in which we 
show relationship between the data by means 
of an equation. 

In this chapter we shall study merely the first two ; 
that is, we shall learn how to make and interpret statistical 
tables and graphs. Let us study graphic methods first. 

I. THE GRAPHIC METHOD OF REPRESENTING 
QUANTITIES 

Section 47. How the Bar Graph is used to picture data. 
A very common method of representing statistics in news- 
papers and magazines, in business offices, and in school text- 



u8 Fundamentals of High School Mathematics 

books, is by means of BAR GRAPHS. Figure 91 illustrates 
the use of the method. It shows by a diagram the marks 
that were given to 37 pupils on an examination in mathe- 
matics. The marks were distributed as follows : 



2 pupils got A 
9 pupils got B 
14 pupils got C 



Table 3 



8 pupils got D 
4 pupils got F 

(that is, they failed) 



ID 




















to 








•&w 














tf. 




































. 




O 5 






























• 





















































































F B C B A 

Fig. 91. Marks received by pupils. 

The teacher, Miss Evans, in sending the marks to the 
principal, represented them in two ways '■ — first by a graph 
like Fig. 91 and also arranged in a table like Table 4. 

Table 4 



Marks received by 
37 pupils in a 
mathematics class 


Date: May 27. 1919 


No. of pupils who 
received each mark 


A. 


2 


B 


3 


C 


1-* 


D 


8 


F 


4 


Total no. of pupils (N) = 37 



How to Represent and Compare Quantities* 119 

Notice carefully how she graphed these facts. First she 
marked off five points on a horizontal line or scale, to 
represent the marks, A, B, C, D, F. Next she marked off 
a number of units on the vertical scale which shows the 
number of pupils who got each mark. Using these two 
scales, and the units which she selected, she erected lines 
at A, B, C, D, F tall enough to represent the number of 
pupils who were given each mark. For example/ since 14 
pupils got C, the line at C is 14 units long ; similarly, for 
A, B, D, and F. 

To make the pictorial effect clearer it is the custom to 
blacken the lines so that they look as in Fig. 92. 



15 

'§•10 

a 




^m 



F P C B A 

Fig. 92. Number of pupils who received each mark, A, B, C, D, and F. 



One of the other teachers in the school suggested that 
the scale need not actually be drawn at the left with the 
units marked off on it. She said the graph would be 
easier to read and would tell exactly the same thing if it 
were made like Fig. 93. 

What do you think ? What represents the number of 
pupils in each case ? 

Figures 92 to 95 give illustrations of several kinds of ver- 
tical bar graphs. It is very important in reading maga- 



120 Fundamentals of High School Mathematics 

14 




Fig. 93. Marks received by pupils. 

zines, newspapers, etc., to be able to interpret accurately 
the meaning of various kinds of graphs. Several are 
included in this chapter, to give you practice in doing 
this. Note that they are taken from many kinds of 
statistics. 

DEATHS from TYPHOID 

KANSAS TEST, 191Z-1973 

+ FLY SEASON-* 

Jan. Feb. MarApr.MayJun. Jul.AugiSep.Oct.NovDec. 

60 
50 
40 
30 
20 
10 


Courtesy of International Harvester Co. 
Fig. 94 

. Interpret Fig. 94. What does the number above each 
bar mean? What has been left off that is included in 



15 


ao 


13 


10 


12 


17 


33 


62 


53 


-W 


38 


27 



How to Represent and Compare Quantities 121 

Fig. 92 ? Can you suggest better ways of graphing such 
facts as these ? 

Note another kind of graph and kind of statistics in 
Fig. 95. What is the meaning of the decreasing heights 
of the bars ? 



100 



97 



83 



53 



33 



21 



12 



4 

jzla 



12 13 14 15 16 17 18 19 20 

Fig. 95. Columns represent number of pupils among each hundred 
beginners who remain in school at each age from 12 to 20. 



Section 48. The use of HORIZONTAL bar graphs to repre- 
sent quantities. In each of the illustrations which you 
have been studying, Figs. 91 to 95, the number of pupils 
or the number of deaths, etc., was represented by vertical 
lines. This is generally done. Sometimes, however, the 



122 Fundamentals of High School Mathematics 



number is recorded on the horizonal axis. Figures 96 to 
100 show how horizontal bar graphs are used to represent 
quantities. , 



.5 t 1.5 7 

T i i i i I i i i i T i i i i I 



2.5 

I T i 



3 

A. 



3.5 

-lJ 



The World 
United States 
Russia 
France 
Austria-Hungary 

Fig. 96. The world's average production of wheat 1905-1909, in billions 

of bushels. 

How could the interpretation of Fig. 96 be made clearer ? 
What do you think about making the " scale " in this form ? 
What improvement on it do you see in the next figure, 
No. 97? 

Total Value of 
PRINCIPAL FARM CROPS JN U.S. 

1,700,000,000 




800,000,000 
797,000,000 
610,000,000 
440,000,000 

U.S.CENSUS,/9I3 
DEPT. OF AGRICULTURE 

Courtesy of International Harvester Co. 
Fig. 97 



How to Represent and Compare Quantities 123 

PERSONS ENGAGED IN 
GAINFUL OCCUPATIONS 

TOTAL in U.S. 2, J, 3 00, 00 o 

10, 500,000 



AGRICULTURE 
MFG.&MECH. 

DOMESTIC 

TRADES-TRANS. 

PROFESSIONAL 




5^00000 
4,800,000 




1,300,000 

Fig. 98 



36% 
24% 

20% 

16% 

4% 

U.S.CENSUSJJOO 



Interpret Fig. 99 as fully as you can. What is the 
purpose of making part of the bar graph black and leaving 
part just in outline ? 

GRAIN FARMING REDUCES PROFITS 

20 YEARS EXP. ILUNO/S , 1Q6Q TO /907 





Cost to 
^row 

25 "bu. 


Bushels 
projit 

m 4 

23 


Totalyleldin 
'bushels 


Corn 


29 
80 


every 




year 


25 bu. 




Corn. 


and 






oats 


25 tu. 


55 




Corn, 


oats and 






clover 









/LUN01S 5ULLET/N /2S 



Courtesy of International Harvester Co. 



Fig. 99 



124 Fundamentals of High School Mathematics 



In what way is Fig. 100 less clearly drawn than Fig. 
99 ? What is the meaning of the * opposite 1914 in 
Fig. 100 ? 

PROGRESS REPORT 

COLLEGES AND HIGH SCHOOLS 
TEACHING AGRICULTURE 



1908 


513 

— 1 

513 ! 236 INCREASE 


1489 


Total 
513 






13 10 


: 1 


809 


809 . 1551 








1 912 


! 1 


2360 


2360 








*I9I4- 


i 









* Estimated 



Courtesy of International Harvester Co. 
Fig. 100 



Section 49. How a continuous line graph can be used to 
represent quantities. Sometimes it is convenient to repre- 
sent the data by a continuous line which joins the top 



lOO' 
90 c 
8tf 

7o c 



7 8 3 



IO 11 12 1 
Time 



3 ■* 



Fig. 101. Vertical lines to represent temperature at different hours of 
the day. (Data in Table 5.) 



How to Represent and Compare Quantities 125 

points of the bars or lines in figures like Figs. 102, 103, 
and 104. For example, suppose you wished to tell some 
one what the temperature was in Chicago at each hour of 
a certain day from 6 a.m. to 6 p.m. It could be done as in 
Fig. 101. 

The graph can be read even more easily, however, if it 
appears as a LINE GRAPH like Fig. 102. 



™» 1 ' ! ' I 




100 1 1 










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, 1 L 


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p 70 . — ■ 








I 




1 




jsr? \ M 




60 : : 1 





6789 10 11 12 1 23456 

TIME 

Fig. 102. Graphic representation of temperatures at various hours 
of the day. 

Note that the horizontal line, or scale, shows the hours. 
This is the time scale. Each large space represents one 
hour. The vertical line, or scale, shows the temperature ; 
on this scale each large space represents 10°, or each small 
space represents 2°. Suppose we wanted to read from the 
graph what the temperature was at 11 o'clock. We find it 
by looking along the time line, or time axis, until we come to 
the point marked 11 o'clock. We then look up to the line 
of the graph. In this case we have to go up to a point 
\\\ small spaces above the 11 o'clock point. By looking 
back, to the left, to the vertical or temperature scale, we 
see that any point on this horizontal line stands for 83°. 
Hence the graph shows that at 11 o'clock the temperature 
was 83°. 



126 Fundamentals of High School Mathematics 

Section 50. A continuous line graph makes it easier to 
INTERPOLATE : to find values between those that are ac- 
tually measured and recorded. Do you see any other real 
difference between these two kinds of graphs, the contin- 
uous line graph in Fig. 102 and the bar diagram of Fig. 101 ? 
What kinds of facts can you get from the graph of Fig. 102 
that you cannot get as easily from Table 5? How can you 
tell about what the temperature was at 8.30 a.m. ? Could 
you tell from Fig. 101 ? What would you have to do to 
Fig. 102 in order to tell ? 

II. THE TABULAR METHOD: MAKING A TABLE WHICH 
WILL REPRESENT THE QUANTITIES 

Section 51. There is a second way in which such facts 
as those in Fig. 102 can be conveniently represented. This 
is by making a table like Table 5. 

Table 5 

an illustration to show the way to tabulate temperature at 

different hours of the day 









A.M. 








3 


P.M. 






Hour 


6 


7 


8 


9 


io 


31 


12 


1 


2 


3 


4 


5 


G 


Temperature 


70 


72 


72 


76 


80 


83 


85 


90 


96 


94 


66 


76 


77 



This method, which we shall call the TABULAR METHOD, 
shows the way the temperature changes at different hours 
of the day. To understand the table, however, requires 
much more effort on the part of the reader than is re- 
quired to understand the first method, which is shown 
above. This pictorial or graphic method shows all that 
the tabular method shows, and has the advantage of being 
more easily interpreted. 

The following questions will help you compare the 



How to Represent and Compare Quantities 127 

graphic and tabular methods of representing the relation 
between two numbers. 

EXERCISE 47 

In order to answer each of these questions, refer to the 
data of Table 5 and Fig. 102. 

1. Find, both from the table and from the graph, 
Fig. 102, the highest temperature. 

2. What was the lowest temperature ? Which 
shows this the more easily, the table or the 
graph ? 

3. Between what hours did the temperature change 
the most rapidly ? 

4. About what do you think the temperature was 
at 9.30 a.m. ? 

5. Between what hours did the temperature change 
the least ? 

6. What might explain the rapid fall in temperature 
between 4 p.m. and 5 p.m. ? 

After answering the questions, are you not convinced 
that the graphic method gives the information which the 
reader may desire much more quickly and easily than the 
tabular method? The fact that this is true has brought 
about a very wide use of graphic methods in all kinds 
of business and industry. Nearly every newspaper and 
magazine contains "graphs" of some kind. Your teacher 
will be glad to have you bring to class any graphs you may 
find in the newspapers or magazines. 

In the following exercise you will get practice in graph- 
ing various kinds of statistical facts. 



128 Fundamentals of High School Mathematics 



EXERCISE 48 



1. 



2. 



PRACTICE IN REPRESENTING FACTS GRAPHICALLY 

Make a bar graph representing the cost of 
education per pupil in Grand Rapids for the 
years 1906-1916. 



1906 


$38 


1910 


146 


1914 


851 


1907 


39 


1911 


49 


1915 


53 


1908 


44 


1912 


49 


1916 


5E> 


1909 


45 


1913 


50 







Represent graphically the results of a test 
which a teacher gave in freshman English. In 
her class of 29 pupils, 4 failed (F), 12 got C, 
5 got D, 6 got B, and 2 got A. 
In a newspaper the graph shown in Fig. 103 was 
printed. It gives the prices of wheat, per bushel, 
from August 5 to August 10. What was the 
price on August 5? On August 7? On August 9? 





*pI/3U 






^" s " ^ 


«"•** ^ 


H I30 J Sr- 


O y^ S 




K 1 


\M 7 


120 J- 


/ 


++" 






1.10 



5 6 7 8 9 10 

AUGUST 

Fig. 103. Graphic representation of prices of 

wheat on various days of August, 1916. 

4. When was the price the highest ? the lowest ? 
Between what dates did the price change most ? 
change least ? 



How to Represent and Compare Quantities 129 



5. 



6. 



7. 



8. 



Each large space on the vertical scale repre- 
sents how many cents ? What is measured 
along the horizontal scale ? What is the unit 
used on this scale ? 

In 1900 40.5% of the total population of the 
United States were living in cities of 2500 or 
more. In 1910, 16.3 % of the inhabitants lived 
in such cities. Represent these facts by means 
of a bar graph. 

Draw a line graph to show the increase in size 
of the graduating class of a high school whose 
graduating classes were as follows : 

1908 56 1911 79 1911 110 1917 131 

1909 72 1912 81 1915 117 1918 139 

1910 73 1913 101 1916 121 

The table below gives the earnings of a book 
agent for the latter part of July, 1915. Show 
the same thing graphically. 



Table 6 



Date 



19 20 21 22 23 24 25 26 27 28 



Earnings -$ p-ool^sol^oolsool^ook 50 ! 3 - 00 ! 8 - 00 !^ 50 ! 9 - 00 



Suggestion. Represent time on the horizontal scale, and earn- 
ings on vertical scale. 

SUMMARY OF IMPORTANT ASPECTS OF GRAPHIC 
REPRESENTATION 

Section 52. In the study of the previous examples, the 
following important aspects of graphic representation 
should be noted : 

1. Graphs always show the relation between two 
changing quantities ; for example, they show 



130 Fundamentals of High School Mathematics 

the relation between the temperature and the 
time of day. 

2. Two rectangular axes are drawn. One of the 
changing quantities is measured on the hori- 
zontal axis ; the other changing quantity is 
measured on the vertical axis. 

3. These axes, or reference lines, are scales, marked 
off in a series of units. Thus, as in our illus- 
trative examples, the horizontal axis may be a 
time scale, marked off into units of one hour 
each, and the vertical axis may be a temperature 
scale, marked off into units of two degrees 
each. 

4. In making a graph one- must choose units very 
carefully in order to be able to get all the 
information on the graph, and yet make it 
stand out as clearly as possible. 

HOW TO COMPARE SUCCESSIVE RELATED FACTS 

Section 53. How graphs are used to compare successive 
or related facts. There are very practical applications of 
the sort of graphing work that you have been studying. 
An illustration is found in graphing the progress that 
pupils make in school. The following example illus- 
trates it. 

Illustrative example. The pupils in a certain mathe- 
matics class took a three-minute practice test every day. 
The object was to see how many examples in evaluation 
(like those in Chapter III) each pupil could solve correctly 
in the three-minute period. The scores of one of the 



How to Represent and Compare Quantities 131 

pupils, Frank Johnson, for fourteen successive days, were 
as follows : 









Table 7 








Jan. 


14th 


1 example 




Jan 


22d 


8 examples 


Jan. 


15th 


5 examples 




Jan 


23d 


10 examples 


Jan. 


16th 


7 examples 




Jan. 


24th 


10 examples 


Jan. 


17th 


8 examples 




Jan. 


25th 


9 examples 


Jan. 


18th 


6 examples 




Jan. 


28th 


11 examples 


Jan. 


21st 


9 examples 




Jan. 


29th 


11 examples 



He made a blank form like that in Fig. 104, and each day 
plotted a point to represent graphically his score. He 
also connected the points that he obtained by a continuous 
line graph, as in Fig. 104. 



12 

a » 



£2 



















ON 


TH1 


bTEi 


ST 














>i'5 


$cd 


REH 




















U l 


is ^ 


/ 








r -' 










V'l 




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/ 






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7 


k 

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* 








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Jan. H 15 J6 J 7 J8 2 J 22 23 24 25 28 29 

Days on "which, the test was taken. 

Fig. 104. Comparison of two pupils' work in a mathematics test on 12 
successive school days. 



How would you interpret this graph ? Tell all the im- 
portant things you can see in it about Frank's ability to 
solve simple equations. What would explain the drop in 
the graph on January 18th, 22d, and 25th ? What do you 



132 Fundamentals of High School Mathematics 



think was true of Frank's skill in doing equations by 
January 29th ? 

Carl Hanley's scores on the same days, in the same 
class, are given in the same chart, Fig. 104. Compare his 
ability to solve equations rapidly and accurately with that 
of Frank. Why do you think his curve rises rapidly on 
January 24th and after ? Interpret the graph carefully. 

Figures 105 and 106 illustrate the use of the line graph. 

19OO'01 *02 '03 '04 '05 '06 «07 '08 *09 'lO *11 '12 '13 

21 
20 
19 
18 
17 
16 
15 
14 
13 

Courtesy of International Harvester Co. 

Fig. 105. Death rate in New York City compared with that in rural 
New York in 14 successive years. 

Why does New York City's curve fall off while rural 
New York's remains fairly level ? 

What is the significance of the rising and falling lines ? 
What do vertical distances represent in this figure?' 
What do horizontal distances represent ? 

Can you make a single table which would show all of the 
facts of Fig. 104 ? How many columns will be required ? 

Use the blank form in Fig. 107 to plot your own score 
on the practice exercise on page 48. 





























20. e 


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20./ 






















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/ 




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15.5 


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5.2 1 


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13.9 






















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How to Represent and Compare Quantities 133 



GRAIN FARMING ROBS THE SOIL 

RESULTS OBTAINED IN A 
BU5HEL5 16 YEARS' TEST IN TENNESSEE 
OF CORN 169 Z to 1907 



80 
70 
60 

50 
40 
30 
20 
10 
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Courtesy of 
International Haroester Co. 



TEWNESSBE BULLET/H 79 

Fig. 106. Comparison of number of bushels of corn obtained per acre under 
live stock farming with the number obtained under grain farming. 



Fig. 107. Plot your scores here. 



134 Fundamentals of High School Mathematics 

HOW TO ARRANGE AND COMPARE SCATTERED DATA 
I. PREPARING A TABLE 

Section 54. How to arrange data systematically in a 
table. It is very important to be able to tabulate facts 

Table 8 



Miss Hitchcock's Class 


Mr. Davis's Class 


Pupils 


No. ex. right 
in 3 minutes 


Pupils 


No. ex. right 
in 3 minutes 


Adams, Ada . . . 
Albright, J. H. . . 
Bass, Dan * . . 










17 
11 
13 


Barwick, Dorothy . 
Barton, John . . . 
Brown, William . 








3 
15 

13 


Biownell, Bessie 
Carlson, Anna 










10 

IS 


Bryant, George . 
Bruner, C. H. . . 








19 
IT 


Crowther, Jas. 










4 


Darling, Jane . . 








14 


Dawes, Janette . 










9 


Dennison, Chas. . . 








7 


Evans, Isabel . . 










11 


Erby, Claude . . 








9 


Finch, Geo. . . 










12 


Evers, Charlotte . 








14 


Ford, Wm. . . 
Harris, David 










11 
9 


Fitzpatrick, J. M. 
Hafifner, Henry . 








15 
5 


Herrick, H. E. . 










8 


Haffher, Margaret 








10 


Hogan, John . . 
Johnson, Emma . 
Lanterman, Anne 










6 
19 
16 


Hall, Marion . . 
Irwin, Jean . . 
Isaacs, Walter 








14 
20 
16 


Lowenthal, Louis 
Manning, Fred . 
Marston, Mary . 
McMurray, Mabel 
Mendenhall, Carl 










15 
10 
11 
12 
15 


Kerby, Harold . 
Lowe, Mary . . 
Macpherson, Edythe 
Marcy, Leslie . . 
Morton, Aaron 








11 

14 
13 

12 

8 


Metz, Pauline 










14 


Norton, Beatrice . 








6 


Owens, Edward . 










12 


Parker, Suzanne . 








5 


Ranney, Geo. 
Reed, Katherine 
Smith, John . . 










5 
3 
14 


Parsons, Julia 
Reynolds, Oscar . 
Rhoads, J. E. . . 








13 
14 
15 


Wright, Evelyn . 










13 


Rice, Marie . . . 






4 


Wright, Betty 










11 


Royston, Ralph . . 






2 



clearly and systematically. You have already had a little 
practice in tabulating quantities which change together. 



How to Represent and Compare Quantities 135 

There are many occasions in which we wish to compare 
scattered data. In such it is necessary to arrange the 
data in a table. An illustration can be given. 

Illustrative example. Miss Hitchcock's class and Mr. 
Davis's class both took a simple equations test. The 
pupils in the two classes made the scores shown in Table 8 
on the opposite page. 

Which class did better work ? Why is it difficult to 
compare the work done by the two classes ? Is it not 
because the test results are not arranged in orderly fashion ? 
For example, in which class is the pupil who got the very 
best scores ? the very poorest ? Were there more very 
good scores in Miss Hitchcock's class or in Mr. Davis's ? 
To answer such questions what do you have to do, as the 
facts are arranged in the above lists ? 

Now note how much more easily you can answer 
the questions from Table 9. The children and their 
scores are now arranged in an orderly way. They 
are arranged so that the pupil who made the best score 
is put first, the next best second, and so on through 
the list. The poorest score is put last. This is called 
RANKING the pupils' scores, or placing them in RANK 
ORDER. 

From a glance at Table 9 on page 136 you can see 
that the best pupil in Miss Hitchcock's class got 19, 
while the best in Mr. Davis's got 20. On the other 
hand, the poorest pupil, 2, is in Mr. Davis's class. 
The pupils are now arranged in convenient order in the< 
two classes. 

Importance of grouping similar scores in a FREQUENCY 
TABLE. Even now, though, it is difficult to tell which 
did the superior work. The figures are still too scattered. 



136 Fundamentals of High School Mathematics 



Table 9 



THE SCORES OF THE PUPILS IN THE TWO CLASSES ARRANGED IN RANK 

ORDER 



Miss Hitchcock's Class 


Mr. Davis's Class 


Pupils 


Scores 


Pupils 


Scores 


Johnson, Emma . 
Carlson, Anna 
Adams, Ada . , 
Lanterman, Anne 
Lowenthal, Louis 
Mendenhall, Carl . 










19 

IS 

IT 

16 

15 

15 

14 

14 

13 

13 

. 12 

12 

12 

11 

11 

11 

11 

11 

10 

10 

9 

9 

8 

6 

5 

4 

3 


Irwin, Jean . . . 
Bryant, George . 
Bruner, C. H. 
Isaacs, Walter 
Fitzpatrick, J. M. 
Rhoads, J. E. . . 








20 
19 
17 
16 
15 
15 


Metz, Pauline 
Smith, John . . 
Bass, Dan . . . 
Wright, Evelyn . 
Finch, George 
McMurray, Mabel 
Owens, Edward . 
Albright, J. H. . 
Evans, Isabel . . 
Ford, Wm. . . 
Marston, Mary . 
Wright, Betty 
Brownell, Bessie 
Manning, Fred . 
Dawes, Janette . 
Harris, David 
Herrick, H. E. . 
Hogan, John . . 
Ranney, Geo. 
Crowther, Jas. 
Reed, Katherine . 


Barton, John . . 
Lowe, Mary . . 
Reynolds, Oscar . 
Darling, Jane . 
Evers, Charlotte . 
Hall, Marion . . 
Macpherson, Edythe 
Parsons, Julia . . 
Brown, Wm. . . 
Marcy, Leslie . . 
Kerby, Harold 
Haffner, Margaret 
Erby, Claude . . 
Morton, Aaron 
Dennison, Chas. . 
Norton, Beatrice . 
Parker, Suzanne . 
Haffner, Henry . 
Rice, Marie . . 
Barwick, Dorothy 
Royston, Ralph . 






15 

14 

14 

14 

14 

14 

13 

13 

13 

12 

11 

10 

9 

8 

7 

6 

5 

5 

4 

3 

2 


Total . . 










zi pupils 


Total . . 








27 pupils 



l Is there any way we can further combine or group them 
to help us decide more easily ? Is there any single score 
in one class that we can compare with the same score in 
the other ? For example, how many pupils did 11 prob- 
lems ? 5 in Miss Hitchcock's class and 1 in Mr. Davis's. 



How to Represent and Compare Quantities 137 

On the other hand, 5 did 14 right in Mr. Davis's and only 
2 in Miss Hitchcock's. 

This shows us that we can save time and make the com- 
parison clearer by grouping together all pupils who did the 
same number of examples. Furthermore, we do not need 
the names of the pupils, to make our comparison. Table 10 
shows the same data grouped in convenient form. 

Table 10 



Test Scores Made by 


No. of Pupils Who Made Each 


Score 


Pupils 


In Miss H's Class 


In Mr 


D's Class 


20 






1 


19 


1 




1 


18 


1 







17 


1 




1 


16 


1 




2 


15 


2 




3 


14 


2 




5 


13 


2 




2 


12 


3 






11 


5 






10 


2 






9 


2 






8 


1 






7 








6 


1 






5 


1 






4 


1 






3 


1 






2 









The table gives the frequency with which each score, i.e. 
each number of examples, was made. For this reason, it 
is called a FREQUENCY TABLE. Notice that the scores 
are still arranged conveniently in RANK ORDER. 



138 Fundamentals of High School Mathematics 



Can you tell any more easily, now, which class did the 
better work on the test ? Is there any single score that we 
could use to compare the work of the two classes ? How 
would it do to compare the number in the two classes who 
solved 11 problems? If we do, Miss Hitchcock's class is 
best, for 5 in her class did 11 problems, and only 1 in the 
other class. Suppose we took 14 problems — then Mr. 
Davis's is best, 5 pupils making this score in his class, 
against 2 in Miss Hitchcock's. 

Bar graphs make the comparison clearer. A clearer com- 
parison can be made if we make a graph of each set of 
scores. Fig. 108 shows this method. 

1* 

I 



1 
















1 


1 1 


llll 


llll ill 


Mi 



1 234567 8 9 lO 11 12 13 14151617 18 19 20 

Test scores made by Miss Hs. class 



lllllllll 



ni 



Fig. 108. 



1 234567 8 9 lO 11 12 13 1415 1617 1819 20 

Test scores made by Mr. D's. class 

Graphic comparison of scores made by two classes 
in a mathematics test. 



Can you tell now which of the two classes is superior ? 
Why do you think Mr. Davis's class is ? Do the scores 
seem to be higher, as a general rule ; that is, are they dis- 
tributed farther along toward the right end of the scale ? 



How to Represent and Compare Quantities 139 

How does the height of the bars in the graphs help you 
decide ? Does the tall bar at 14 examples in Mr. Davis's 
class influence your decision ? 

COMPARING DISTRIBUTIONS OF DATA BY MEANS OF 
SINGLE NUMBERS 

Section 55. The use of a single number: The mode, the 
median, and the average. When we work with data that 
are so scattered and have to do with so many different 
values, we need single numbers which will tell us the 
tendency of the data to center around particular values. 
This is called the CENTRAL TENDENCY of the data. Thus, 
we are interested to find whether there are numbers that 
represent the central tendency of all the measures. There 
are three numbers that do this very well : (1) the mode or 
commonest number, and (2) the median or middle number, 
and (3) the average. We shall discuss these next. 

Section 56. First : THE MODE : the commonest number. 
This tallest bar at 14 examples, in the scores of Mr. Davis's 
class, tells you that the score, 14, was made more fre- 
quently than any other. It is the commonest score. People 
who work with scattered statistics call this the MODE (the 
word comes from the French word, " la mode" meaning 
" the fashion "). 

The mode of the scores made by Miss Hitchcock's class 
is 11. This is 3 examples less than the mode of the scores 
in Mr. Davis's class, and tends to show that the latter is 
superior. What is the mode of the marks made by the 
pupils represented in Fig. 91 ? What of the data in 
Fig. 94? What is the mode in Figs. 95 and 98? Just 
what does this one figure tell you about each graph or each 
distribution ? 



140 Fundamentals of High School Mathematics 

Do you think this single number represents the tendency 
of the measures to be gathered at one place on the scale ? 
Thus one can use the mode to help compare two sets of 
data. The mode is the easiest single number to find, for 
you simply determine from the tallest bar of the graph 
which number occurred most frequently. 

Section 57. Second : THE MEDIAN : the middle number. 
There is another number that is easily found and that 
represents all of the numbers in the list rather we'll. This 
is the middle number. It is called THE MEDIAN. It can 
be found by merely counting from one end of the list. 
Why ? For example, the median score made by Miss 
Hitchcock's class is 11, because there are 27 pupils in the 
class and the middle one is the thirteenth. This thirteenth 
pupil from either end is one of the 5 pupils who made 
scores of 11. So the median score is 11. The median in 
Mr. Davis's class is 14. Why ? How does this help you 
to decide which class did the better work ? All of our 
work so far shows that Mr. Davis's class did. 

Section 58. Third : THE AVERAGE : another single num- 
ber which represents all of the data. In the elementary 
school you have already learned how to find the average of 
a series of numbers. For example, how do you find the 
average of the scores of the 27 pupils in Miss Hitchcock's 
class ? of those in Mr. Davis's class ? What do you total ? 
What do you divide by ? If you let each fact or measure 
be represented by m, and the total number of measures 
by n, can you make a formula for the value of the 
average ? What is the average in each case ? Which is 
the higher ? What does this tell you about the work of 
the two classes ? How does it compare with what the 
median and the mode each told you ? 



How to Represent and Compare Quantities 141 

The median and the mode and the average are measures 
of " central tendency." So we can say that the most 
typical or representative score in Mr. Davis's class was 13, 
while in Miss Hitchcock's class it was 11. 

What is the median of the distribution in Fig. 93 ? 
What is the average in Fig. 108 ? In what way are these 
measures of central tendency? 

THE GENERAL SHAPE OF THE DISTRIBUTION OF MOST HUMAN TRAITS 

Section 59. How scores are generally distributed. Turn 
back to Fig. 91. Study the frequency with which each 
mark, A, B, C, D, and F, occurred ; this frequency gives a 
graph, or curve, of a particular shape. Now study the 
shape of the graphs in Fig. 108. Do you notice any 
resemblance between them ? 

























FABLE 1-c 
















Heights in. 
centimeters 


(J) c 
(N ( 

1— < p 


O 
3 S 


CM 

to 

p—t 


t 


ti> 

rO £ 


3 O r 

it: 




!S§ 


to 10 u 


D OQ 
T 10 

-t i-i 









Frequency, 
i.e., no. boys 


M I 


$ 


X 


rO 


So 


c 

i-l r 


3 N t 

3 c 


00 to 
^ 00 w 


vD tO ( 
fO (N r 


% "0 


f-l 




as 


[n Table 11, the heights of 12-year-old 
they were actually measured in a certair 


dovs are gi 

1 school. • 


ven 

rhe 



































































































































































































































































































































































































































































































































































































































































































































































































































Fig. 109 



142 Fundamentals of High School Mathematics 

heights are measured in centimeters. Plot a graph in the 
blank form in Fig. 109 to represent these facts. 

In exactly what way does this graph resemble the others ? 
In Figs. 110 to 113 measurements of four other human 
traits are graphed. What fact do you notice of the shape 
of these graphs that is like the others ? 

B 

2, 
Sl200 

f—i 

|900 

I 

.6 600 

I 

&300 



£ 58 60 62 64 66 68 70 72 74 76 78 

3tature in inches 
Fig. 110. Heights of 8585 men grouped in inch intervals. 



m 



Fig. 111. How 904 children distributed in intelligence 
from very bright to very dull. 

Section 60. These graphs all resemble each other in one 
way. The heights of the lines or bars at the extreme low 
and high ends of the curve are very short. Those nearer 



How to Represent and Compare Quantities 143 



504-3 



■4462 



3536 



2878 



1373 



S3 



3536 



1884 



527 



5 15 25 35 -45 55 65 75 85 95 105 115 125 

Fig. 112. Number of pupils writing at each speed from 
to 9 letters per minute to 120 to 129 letters per minute. 
Data for 25,387 pupils in four upper grades. 



a, 

i 

i 



r 

*5 




































































15- 

IO- 

5- 



























































































12 



Fig. 



2 54 5 6 7 8 9 10 U 

Number of problems solved 
113. Number of pupils who solved various numbers 
of algebra problems. 



J 5 



144 Fundamentals of High School Mathematics 

the middle are longer and that nearest the very middle 
tends to be longest. We interpret this fact by saying 
that there are always more average or middle measures 
than extreme ones. Notice how true it is of every graph 
of facts that have, to do with human beings. More pupils 
got average marks (Fig. 91) than very good or very 
poor. There were more boys of average height (that is, 
about 142 centimeters) than there were very tall or very 
short (Fig. 110). More fifth-grade pupils in Cleveland 
could write at an average rate of about 75 words a minute 
than at a very rapid rate, like 125 words, or a very slow 
one, like 10 to 25 words per minute. More pupils could 
solve about 8 examples in algebra than any other number. 
So it is with all measurements of human traits. Many 
careful measurements have been made of both physical 
and mental abilities during the past 100 years and they all 
show the same kind of graph. 

Section 61. You have seen that there are always more 
" average " measures than extreme ones. We have already 
used single numbers to characterize this central tendency. 
Now we see that the graphic representation of the facts 
confirms the conclusions which we obtained by comparing 
statistics by single numbers. 

SUMMARY OF CHAPTER 

The important points of this chapter are as follows : 

1. Facts of similar kind are called statistics. 

2. There are three methods of dealing with quan- 
tities in mathematics: (1) the tabular method; 
(2) the graphic method; (3) the equational or 
formula method. 



How to Represent and Compare Quantities 145 

3. Horizontal and vertical BAR GRAPHS are used to 
picture statistical facts. 

4. Continuous line graphs are also used and enable 
us to INTERPOLATE. There is an important 
summary of graphic methods on page 129. 

5. Statistical facts can also be represented by tabu- 
lating them in a table. It is not so easy to com- 
pare them by this method as by the graphic 
method. 

6. Line graphs are used to compare successive facts. 

7. Scattered statistical facts can be grouped and 
compared clearly by arranging them in RANK 
ORDER. 

8. It is still clearer to group the quantities in a 
FREQUENCY TABLE. 

9. The frequencies may be clearly shown by plot- 
ting them as vertical or horizontal bar graphs. 

10. When we plot such FREQUENCY DISTRIBU- 
TIONS the bars commonly are longest (that is, 
the frequencies are greatest) near the middle or 
average part of the scale. 

11. Hence we can compare two or more frequency 
distributions by computing an average to show 
the CENTRAL TENDENCY. 

12. There are three important measures of CENTRAL 
TENDENCY: (1) the MODE, i.e., the most fre- 
quent number; (2) the MEDIAN, i.e., the middle 
number; (3) the AVERAGE, i.e., the sum of the 
measures divided by the number of measures. 



146 Fundamentals of High School Mathematics 

13. With measurements of traits of human beings, 
the average or central measurements always 
occur most frequently. The measurements occur 
less frequently as they become very small or 
very large. We interpret this as follows : 
" More ordinary people occur in the world than 
unusual ones." 



REVIEW EXERCISE 48 a 

Two classes in a certain school took the same test in 
Simple Equations. The results are given here. 



Mr. Evans's Class 


Miss Brown's Class 


Pupil 


No. 

examples 

right 


Pupil 


' No. 
examples 
right 


A 


1 
4 
5 
5 
3 
6 
4 
3 
5 
4 
3 
7 
2 
5^ 
2 
4 
8 
4 
5 
1 
6 
2 
4 


A 


2 


B 


B 


4 


c 


c 


5 


D 


D 


3 


;E 


E 


4 


F 


F 


5 


•G 


G 


4 


H 


H 


7 


I 




5 


T , 


T 


7 


k::::::::::::::::""::::::::: 


t:::::::::::::::::::::::::.:. 





L 


L 


3 


M 


M 


4 


N 


N 

O 


2 


O 


4 


p 


p 


4 


O 


O 


3 


fi..:.::::.::. :.....::::: :::::: 


it: 


5 


s 


s 


4 


T 


T 


6 


U . 


u 


1 


v 


V 


6 


w... 


w 


6 




X 


2 




Y 


3 









Which class did the better on the test? What is the 
median number of rights in each class ? Find the average. 
Represent the two tables graphically. 



CHAPTER VIII 

HOW TO REPRESENT AND DETERMINE THE RELA- 
TIONSHIP BETWEEN QUANTITIES THAT CHANGE 
TOGETHER 

Section 62. In mathematics we deal with two distinctly- 
different kinds of graphs. In Chapter VII we tabulated and 
graphed data in which there was not necessarily a causal 
relation between the two quantities ; for example, the tem- 
perature and the hour of the day. This is obviously true of 
the mere graphic representation of statistical facts like popu- 
lations, school marks, attendance, wealth, production, etc. 
For this kind of graph no equation or formula can be made. 
We speak of this kind of graph as a STATISTICAL GRAPH. 

Section 63. As we go about our daily work, we commonly 
deal with quantities which change together. For example, 
the cost of a railroad ticket changes as the number of 
miles you travel changes ; that is, the cost and the distance 
change together. Or, the distance traveled by an autoist, 
if he goes at the rate of, say, 20 miles per hour, changes 
as the number of hours which he travels changes ; that is, 
the distance and the time cha?tge together. As a third 
illustration, suppose you wanted to make a trip of 100 miles. 
We know that the time required will change with or be 
determined by the way in which the rate changes ; that 
is, the time and the rate change together. 

Section 64. The fact that we are always dealing with 
situations of this kind makes it necessary for us to know 
how to represent and determine these quantities which 
change together, or which are related in some definite way. 
Mathematics shows us how to describe or express them. 
In fact, it is the chief aim of mathematics to help you to 
see how quantities are related to each other and to help 
you to determine their values. 

i47 



148 Fundamentals of High School Mathematics 

Section 65. The three methods of representing relation- 
ship. You learned in Chapter VII that there were three 
methods of representing quantities, of comparing them, 
and of determining the relationship between them. These 
were: (1) The graphic method, (2) the tabular method, and 
(3) the equational or formula method. In the problems 
just taken up in Chapter VII you used the tabular method 
and the graphic method of representing and comparing 
facts. In those cases there was no relationship expressed 
and hence no possibility of using the equational or formula 
method. 

In this chapter, however, we shall deal with facts which 
are related and learn how to use all three methods of 
expressing and determining the relationship between them. 
A good illustration is that of the relationship between 
(1) the time a railroad train travels and (2) the distance the 
train travels. Let us take an example. 





• 


7 


OT^X S 


250 S 


2 


7 


s 




200 " /* 


~? 


*r 


W >"* 


U.„ • «/ 


£150 7 




£ * 


fc ^ 


to - y 


o 100 - - ~X- 


" ^ 2 ■ 


■/r 


S 


CA S 


50 £_ 


^ 


J? 


s 


r\ < 


0^ _ .__ 



10 



11 12 
TIME 



Fig. 114. The line shows relationship between time spent 
and distance traveled. 



How to Represent Relationship 



149 



Illustrative example. An east-bound train, run- 
ning at 40 miles per hour, left Chicago at 8 a.m. 
Show from the graph, Fig. 114, how far the 
train was from Chicago at 10 a.m. ; at 11 a.m. ; 
at 11.30 a.m. ; at 2 p.m. At what time was the 
train 100 miles from Chicago ? 200 miles ? 
In Fig. 114 how many miles does each small 
space represent ? How many hours does each 
large space equal ? 
Table 12 also shows the relationship between (1) the 
time the train traveled and (2) the distance it went. 

Table 12 



At a given time (hrs.) 


1 


2 


3 


4 


5 


6 


7 


the distance was (mi.) 


40 


do 


120 


160 


200 


240 


280 



Second illustrative example. Another good illustration 
of the graphic method of representing relationship is that 



shown in Fig. 115. 



$2.40 
180 



H 
gl.20 

U 



.60 



.ooEgg 



6 8 10 12 14 

NUMBER OF YARDS 



16 



18 



20 



Fig. 115. The line shows the relationship between the number 
of yards of cloth purchased and the total cost. 



150 Fundamentals of High School Mathematics 

Figure 115 is a price graph which shows the 
cost of any number of yards of cloth at 12 
cents per yard. From it we can find the cost • 
of any number of yards. For example, the cost 
of 15 yd. is found by finding the point on the 
horizontal scale which stands for 15 yd., then 
by finding the point on the cost line directly 
above this point. This appears on the cost line 
as point A. Now, to find the cost of 15 yd. we 
find the point on the cost axis, horizontally 
opposite the point A which already stands for 
15 yd. The cost proves to be $1.80. Thus 
we see that point A stands both for 15 yd. and 
for 11.80. In the same way the point B shows 
that 8 yd. on the horizontal scale corresponds to 
96 ^ on the vertical scale. The point C shows 
tljat 18 yd. on the horizontal scale corresponds 
to $2.26 on the vertical scale. 
Table 13 shows the same relationship but not so clearly. 
Table 13 



If the no . 
of yards is 



10 



11 



12 



13 



the total 
cost is($) 



12 24 .36 



48 



.60 



72.84 



96 



108 1.20 1.32 1.44 L56 



THE FORMULA OR EQUATIONAL METHOD OF EXPRESS- 
ING AND DETERMINING RELATIONSHIP 



US 



Section 66. The picture of relationship. Now let 

apply the formula method to the second illustrative ex- 
ample given in the preceding section. Make a formula for 
the cost of any number of yards of cloth at 12 4 per yard. 



How to Represent Relationship 151 

Note that the graph and the table and this formula which 
you have just made tell exactly the same thing. The graph 
tells the relation between the cost and the number of yards 
purchased more clearly because it presents it to the eye as 
a picture. To tell from the graph the cost of any particular 
number of yards requires only a glance ; to tell from the 
formula or equation, 

C=.12n, 

requires that we substitute some particular value of n in 
the equation and then that we find the value of C. 

EXERCISE 49 
LET US GET MORE PRACTICE IN USING THESE THREE METHODS 

1. Draw a graph showing the price of any number 
of pounds of beans at 9 cents a pound. From 
it find the cost of 5^ pounds ; of 12 pounds. 

2. Now, write a formula which represents the cost 
of any number of pounds at 9 p a pound. Note 
that the graph and the formula tell tlie same thing. 

3. Draw a graph for the cost of a railroad ticket 
at 3 (f, a mile. 

4. If c = .03 m is used as the equation for the cost 
of any railroad ticket at 3 <f, a mile, show that 
by letting m have particular values, such as 2, 
3, 7, 10, etc., we get values for c> from which 
we can make the graph. . 

5. A number of rectangles have the same base, 
5 in. Write an equation for the area of any 
rectangle which has a 5-inch base. (Use h for 
the altitude, or height.) 



152 Fundamentals of High School Mathematics 

6. Draw a graph for the area of any rectangle 
whose base is 5 in. by using the equation you 
got in Example 8. (Let h have particular 
values, such as 2, 3, 4, 7, 10, and find the cor- 
responding area, in each case.) 

7. A west-bound train leaves Chicago at 7 a.m., 
going 30 miles per hour. Show graphically its 
progress until 4 p.m. 

& Using d = 30 t for the equation of the train in 
Example 7, show that the graph could have 
been made from the results obtained by letting / 
have particular values. 

9. The movement of a train is described by the 
equation d = 25 t. Draw a graph showing the 
same thing. 

10. A boy joined a club which charged an initiation 
fee of 25 cents. His dues were 10 cents each 
month. Draw a graph to show how much he 
had spent at the end of any number of months. 

11. What formula or equation will represent the 
same thing as the graph in Example 10 ? 

I. VARIABLES : Quantities which are continually varying. 
II. CONSTANTS : Quantities which are always fixed or un- 
changing. 

Section 67. In all the. examples which you have just 
solved graphically there have been changing or varying 
quantities ; for example, in the graph of the motion of 
a train, the distance and the time vary as the train 
moves along its trip ; or in any cost graph the cost varies 



How to Represent Relationship 



153 



(that is, increases and decreases) as the number of articles 
varies. 

But in these examples, some of the quantities do not 
change or vary. To illustrate : the rate of the train (as in 
Example T, 30 miles per hour) remains fixed, or constant, 
as the train moves along ; and the price per unit of any 
article (for example, cloth at 12 ^f per yard) remains fixed 
or constant in any particular example. 

Thus, in any problem we may have two kinds of quanti- 
ties : first, those that change or vary ; and second, those 
that remain fixed or constant. We call them, respectively, 
variables and constants. For example, in the formula for 
the area of any rectangle whose base is 4 units, A = 4:h, 
it is clear that A and h are variables, and that the base, 4, 
remains constant. In other words, if h is 2, then A is 8 ; 
if h is 3, then A is 12 ; if h is 7, then A is 28, etc. Thus, h 
can change, but as it changes, A also changes, since A is 
always 4 times as large as h. Hence, 4 is the " constant " 
in the equation, and A and h are the "variables." Note 
that there is a definite relation between A and h. A is 
always 4 times h. 



EXERCISE 50 



Determine the variables and the constants in each of the 
following examples. ,Give reasons for each decision that 
you make. 



1. c = 2ttR 

2. d = i0t 

4. x = y + 4 



5. c = 10 m + 25 

6. A=s 2 

7. P=2b + 2/i 

8. c=10m + 50 



154 Fundamentals of High School Mathematics 

GRAPHS SHOW THE RELATION BETWEEN TWO 
VARIABLES 

Section 68. Each of the three methods shows relation- 
ship. A cost graph, such as Fig. 115, really shows the 
relation between the number of units (lb., doz., or yd., etc.) 
purchased and the total price paid. A graph of the move- 
ment of a train {e.g. Fig. 114) which runs at a constant rate 
shows the relation between the number of hours (the time) 
and the number of miles traveled (the distance). Saying 
that these graphs show the relation between the numbers 
represented by them means that if we read a particular 
value of the time, such as 2 hr. or 5 hr., we can find 
the number of miles which corresponds to that number 
of hours. Thus, graphs show the relation between two 
variables ; that is, they show the values of one variable 
which correspond respectively to the values of another 
related variable. 

A formula also > shows the relation or connection between 
the two variables. For example, the formula for the area 
of any rectangle with a 3-inch base, which is A — 3 //, shows 
that the value of A must always be three times tJie value ofh, 
or, in other words, the area is always three times the height. 
At first it is more difficult to understand the formula than 
the graph, but as you advance in mathematics the formula 
will become more important and significant. 

Thus, as we stated at the beginning of the chapter, there 
are three methods of showing the relationship or connec- 
tion between the kinds of variables we have studied : 

I. THE TABULAR METHOD 
II. THE GRAPHIC METHOD 
III. THE FORMULA METHOD 



How to Represent Relationship 



155 



Let us illustrate for one example each of these methods. 
A man walks at the rate of 6 miles per hour. Show 
the relation or connection between the distance he 
walks and the number of hours he walks. 

I. Tabular method 

Table 14 



If the no. of hours is 


1 


2 


5 


8 


10 


12 


then the distance is 


6 


12 


30 


48 


60 


72 



The table shows the relationship between the time spent 
and the distance traveled. 



II. Graphic method 







50 










s 




--T +' 




■** 


40 


+? 




- Jt- A 




■** 


. , 


S 




^' 


UoO ^ 




Z X X 




2 - - 








H -T--+- ?* 




to 20 s' 








fi v 




s 




-in S 




10 y> 




t s i - 




^ X 




z it 








0^ 1 ! 





012345678 
TIME 
Fig. 116. The line shows the relationship between the time 
spent and the distance traveled. 



III. Formula method 

The equation shows the relationship between the time spent 
and the distance traveled. 



156 Fundamentals of High School Mathematics 

EXERCISE 51 
PRACTICE IN REPRESENTING THE RELATION BETWEEN VARIABLES 

Show by three methods the relation between the vari- 
ables in the following : 

1. The area of a rectangle whose base is 8 in. and 
its height. 

2. The cost of belonging to a club which charges 
an initiation fee of 50^, and 10^ per month for 
dues. 

3. A freight train leaves Chicago at 10 a.m., at the 
rate of 25 miles per hour ; at 1 p.m. a passenger 
train leaves Chicago, running in the same direc- 
tion, at the rate of 40 miles per hour. Show 
graphically at what time the passenger train will 
overtake the freight train. See Fig. 117 for solu- 
tion. How does the graph show that one train 
will overtake the other? If t represents the 
time of the freight train, what formula will rep- 



w 









250 






















7 * * 






A^ 


200 




>* 






s? 






*Z' 






-*<±*s 


it:r, 




^ A 


150 




.,<!? Z _ _ 


. 










^y y 






^ *.*" 


1 




^^ ,2 


100 






. 




y'' >' 




^^ 


^ 




^ ^ 


• 




^^ 


.?" 


50 


^ 






^^ 


** zt~ ■ ~ 




^. ? * 


_,' 




\** * 




_ ^' 






^ 


:_ _ _ *t 





10 



11 



12 



2 3 

TIME 



Fig. 117. The lines show relationship between the time spent and 
the distance traveled by each train. The point of intersection indi- 
cates the time at which they will meet and how far each travels. 



How to Represent Relationship 157 

resent the distance it travels ? What will repre- 
sent the time the passenger train travels? 
What formula will represent its distance ? 

4. A slow train left Cleveland at 6 a.m., running 
uniformly at the rate of 30 miles per hour. At 
10 a.m. a faster train left Cleveland, running in 
the same direction, at the rate of 40 miles per 
hour. Show graphically at what time the faster 
train will overtake the slower one. 

5. A freight train left St. Louis at 7 p.m., running 30 
miles per hour. At 11.30 p.m. an express train 
started in the same direction. Show graphically 
at what time it will overtake the freight train, if 
it runs 45 miles per hour. 

6. A bicyclist left a certain place at 10 aIm., travel- 
ing at the rate of 8 miles per hour. How long 
would it take a second bicylist to overtake the 
first one, if he travels at the rate of 10 miles per 
hour, and starts 2 hours later ? 

7. An elementary school graduate began work with 
a weekly wage of $ 9, and received an increase 
of 25 ^ every week. One high school graduate 
began work at a weekly wage of only $ 6, but re- 
ceived an increase of 40 ^ every week. Make 
a graph which shows the wage of each at the 
beginning of every week for 20 weeks. From 
the graph tell when they will be receiving the 
same wage. 

Section 69. Two different kinds of data which can be 
graphed. We should distinguish between the two kinds of 
data which we have graphed. 



158 Fundamentals of High School Mathematics 

A. The statistical graph. The first kind includes all 
those for which no formula or equation can be made. 
Recall the first illustrative example in this chapter : the 
relation between the time of day and the temperature. 
Clearly, no formula can be made which will always show 
the relation between the two variables in this kind of ex- 
ample. Thus, there are only two ways to show or repre- 
sent this kind of relation : (1) the tabular method, (2) the 
graphic method. 

B. The mathematical law. The second kind of example 
which we have been graphing is illustrated by any of those 
examples for which we made a formula. For example, we 
have such illustrations as : the graph showing the relation 
between the distance traveled by a train running at 30 miles 
per hour and the time the train travels. This belongs to the 
second kind of graph, because we can make a formula for 
the relation between its variables. The formula is : 

d=sot. 

Thus, there are three ways to show the relation between 
such variables as these : (1) the formula or algebraic 
method, (2) the tabular method, and (3) the graphic 
method. 

In mathematics, we say that the second kind of graph, 
for which an equation can always be made, states alge- 
braic laws, or mathematical laws, because there is always 
a definite relation between the variables. The first kind of 
graph, for which no definite law or equation can be made, 
is sometimes called a statistical graph. It is this kind 
that is most frequently seen in newspapers and magazines. 
In mathematics, however, the other kind, that which states 
" laws," is nearly always used. 

The next exercise will give practice in making both 



How to Represent Relationship 



159 



kinds of graphs. It is important to tell whether the in- 
formation to be graphed (generally called the data) can be 
expressed by an algebraic law or formula. 

EXERCISE 52 

l. The following table shows the average heights 
of boys of different ages. Construct a graph 
showing this information or data. 



.Age in years 


2 


4 


6 


8 


IO 


12 


14 


16 


ia 


20 


Height in feet 


1.6 


2.6 


3.0 


3.5 


4.0 


4.8 


5.2 


5.5 


5.6 


57 



3. 



4. 



5. 



6. 



Represent ages on the horizontal scale. 
When does the average boy grow the most 
rapidly ? the most slowly ? 

Is there an algebraic " law," or formula, which 
shows the relation between these two variables, 
age and height? 

Mr. Smith joined a lodge which charged $25 
initiation fee, and dues of 1 2 per month. Show 
graphically the relation between the cost of be- 
longing and the time one belongs. 
Is there an algebraic "law," or formula, which 
shows the relation between the variables, cost 
and time ? 

The information or data of the following table 
represent the area of a square of varying sides : 
Show this relation between the area of the 
square and its side graphically, using the verti- 



If the side is 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


then the area is 


1 


4 


9 


16 


25 


36 


49 


64 


81 


100 



160 Fundamentals of High School Mathematics 

cal scale to measure areas and the horizontal 
scale to measure sides. 

7. Is there an algebraic " law," or formula, which 
shows the relation between the variables here ? 

8. The weights of a baby boy who weighed 8 lb. 
at birth are given for each month of his first 
year by the table : 



Month 


1 


2 


3 


4 


5 


6 


7 


8 


9 


io 


11 


12 


Weight 


si 


n4 


12 i 


14| 


151 


16! 


18 


19 


19! 


20 


21 


22 



Represent this graphically. 
9. Can you make a formula to show the relation 
between weights and months ? Then what 
kind of graph is this, statistical or algebraic ? 
10. This table shows the cost of various amounts of 
flour. 



If the no. of pounds is 


1 


2 


3 


5 


10 


then the cost is ($) 


.08 


.16 


.24 


.40 


.80 



11. 



12. 



Can you make a formula which shows the rela- 
tion between these variables ? If you should 
graph these variables, what kind of graph would 
you get, a straight line, or a broken line ? 
Each member of a class made problems for the 
others to solve. One boy presented the follow- 
ing table : 



If the small no. is 


4 


7 


IO 


20 


50 


2 


the large no. is 


8 


14 


20 


40 


100 


4 



How to Represent Relationship 



161 



13. 



He wanted to see if anybody in the class could 
make a formula showing the relation between 
the two numbers. (Use / and s for the num- 
bers.) Could they ? 

The next day a more difficult problem was 
given. The object was to make a formula to 
show the relation between the numbers (varia- 
bles) in the following table : 



If fche small no. is 


2 


3 


-4 


5 


io 


the large no. is 


5 


7 


9 


11 


21 



Can you solve the problem ? 

14. Make a graph of the formula for example 13. 

15. Make up a problem similar to examples 13 and 
11 for the members of your class to solve. 

EXERCISE 53 
PRACTICE IN SOLVING EASY EQUATIONS 

Find the value of the unknown in each of the following 
Check each example. 

1. 2^-3 = 14 

2. 12+^ = 3,r 

3. 1^-2 = 18 

4. 3£ + 2£ = 2£ + 21 

5. 5^-2 = 13 

6. 120 = 4;r + 20 

7. 8^-9 = 91 

8. 6* = * + 28 

9. 10j/-9 = 47 
10. 12£ = 30 + 7£ 



11. 


|>=18 


12. 


|^+2 = 10 


13. 


^ + 1^ + 5 = 30 


14. 


f£+V = 26 


15. 


f = f +1 


16. 


-!« = tV + 2 


17. 


\y + %y=y + 15 


18. 


20 = f/ 



1 62 Fundamentals of High School Mathematics 

SUMMARY OF CHAPTER VIII 

This chapter should make clear the following truths 

1. Important facts about quantities, which change 
together, are more easily read and interpreted if 
they are represented graphically. 

2. Graphs always show the relation between two 
varying quantities. 

3. There are three fundamental methods of describ- 
ing the relationship between related variables : 

a. The Graphic Method of expressing " Law " ; 

b. The Tabular Method of expressing " Law" ; 

c. The Formula, or Algebraic Method, of stating 
"Law." 

They tell the same thing, the graphic method 
most clearly. 

4. We deal with two important kinds of quantities : 

(1) Variables, quantities which are continually 
varying ; (2) constants, quantities which are 
fixed or unvarying. 

5. It is important to distinguish between two kinds 
of graphs : (1) Statistical graphs for which no 
law or relationship can be expressed ; and 

(2) graphs of mathematical laws. 

EXERCISE 54 

1. If n represents a boy's present age, state in 
words what the expression n + 7 = 22 means. 

.2. Give a formula for the base of a rectangle when 
the area and height are known. 



How to Represent Relationship 163 

3. Represent the number of cubic yards in a box- 
shaped excavation when the dimensions are ex- 
pressed in feet. 

4. If m, s, and d represent the minuend, subtra- 
hend, and difference respectively, what formula 
will show the relation between these numbers? 

5. Show by a formula the relation between the 
product,/, multiplicand, M, and multiplier, m. 

6. Give the meaning of the formula i=prt. 

7. Divide each side of the formula V= Iwh by Iw 
and tell what the resulting formula means. 

8. Give a formula for the volume of a cube whose 
edge is s. 

9. Evaluate the above formula when s = 3.2. 

10. Translate into words the formula d = rt. 

11. Divide each side of the formula d= rtby r and 
tell what the resulting formula means. 

12. In the formula c — np, n represents the num- 
ber of articles bought, p represents the price 
of each, and c represents the total cost. Trans- 
late it into a word statement. 

13. Divide each side of the formula c — np by n, 
and tell what the resulting formula means. 

14. Does x — 4 satisfy the equation x 1 -f- 6 x — 40 ? 

15. Solve the equation 10 y + 7 = 52 + 4j/. 

16. Is the equation x + y + 3 = 20 satisfied if x = 8 
and y=9? Can you find any other values of 
x and y which will satisfy this equation ? 



164 Fundamentals of High School Mathematics 





22 _ 

J. 
1.5 £ 


5.5 
= 45 


Or) 


X 

12" 


= 60 


w 


.34 = 


= 85 



17. Solve each of the following equations, thinking 
of each example as asking a question : 

x 
(b) .5^ = 17 

(V) .4/ = 80 

(V) ^ = 4.25 

Tell what you do to each side of the equation ; 
that is, tell whether you add, subtract, multiply, 
or divide, on each side. 

REVIEW EXERCISE 53 

1. Write in algebraic language : The volume of a 
sphere is four thirds the cube of the radius 
times 7T. 

2. Which would you rather have, 4:x + by dollars, 
or 5x + ±y dollars, if x = $20 andj/ = $16? 

3. If V is the volume of a cone, b the area of its 
base, and h its height, then 

V=\bh. 

Write this formula in words. 

4. Nurses keep temperature records of fever pa- 
tients. For one patient the following degrees 
of fever were noted : 

2, 5.4, 4, 6.1, 4.5, 6.5 } 5.3, 6.7, 4.5, 6.5, 5.9, 6.2, 
6.9, 5, 6.4, 4.7, 5.8, 7.6. 

These readings were taken an hour apart. 
Show graphically this patient's successive tem- 
peratures. 



How to Represent Relationship 



165 



5. If you know that 

10 a - 7 = 4 a + 35, 
then what do you do to each side to get 

10 a = 4 a + 42 ? 
How do you get 6 # = 42 ? 
Why does a = 7 ? 

Prove that « = 7. 

6. Get the hourly temperature for 24 hours from 
your daily paper, and construct a graph to rep- 
resent the changes in temperature. 

7. The sum of two numbers is 18 and their differ- 
ence is 4. What are the numbers ? 

8. Show that 2 a and cP are unequal by choosing 
some particular value for a, such as a — 6. Do 
you think there is any possible value for a 
which would make 2 a — cP ? 

9. What product is obtained by using 7 as a 
factor twice ? by using 2 a as a factor three 
times ? 

10. The following table shows the number of feet 
required to stop an automobile running at vari- 
ous speeds. 



At a speed of 
(miles per hour) 


IO 


15 


20 


25 


30 


35 


40 


50 


a car should 
stop (feet) 


9.2 


20.8 


37 


53. 


83.3 


104. 


148. 


231. 



Represent this graphically, measuring the speed 
on the horizontal axis. 



CHAPTER IX 

THE USE OF POSITIVE AND NEGATIVE NUMBERS 

Section 70. We need numbers to represent opposite quali- 
ties, or numbers of opposite nature. The examples in the 
following exercises will illustrate what is meant by opposite 
qualities, or numbers of opposite nature. We shall take four 
different kinds of illustrations : (1) opposite numbers on 
a temperature scale, (2) opposite numbers on a distance 
scale, (3) opposite numbers to represent financial situations 
("having" and " owing"), (4) opposite numbers on a time 
scale, to represent " time before " a beginning point and 
•"time after." 

FIRST ILLUSTRATION: OPPOSITE NUMBERS ON A 
TEMPERATURE SCALE 

EXERCISE 56 

1. The top of the mercury column of a thermometer 
stands at zero degrees (0°). During the next 
hour it rises 3°, and the next it rises 4°. What 
is the temperature at the end of the second hour ? 

2. The top of the mercury column stands at 0°. 
During the next hour it falls 3°, and in the next 
it falls 4°. What is the reading at the end of 
the second hour ? 

3. If it starts at 0°, rises 3°, and then falls 4°, what 
is the reading ? 

4. If it starts at 0°, falls 3°, and then rises 4°, what 
is the reading ? 

These examples show that we must distinguish two 
Mnds of temperature readings, (1) those above zero and 
(2) those below zero. People have agreed to call readings 
above zero POSITIVE, and readings below zero NEGATIVE. 

1 66 



The Use of Positive and Negative Numbers 167 

Thus, if the mercury starts at zero and rises 4°, it will be at 
positive 4°, or, more briefly, + 4°. But if it starts at zero 
and falls 4°, it will be at negative 4°, or — 4°. In the re- 
mainder of these examples you should describe the mercury 
readings as positive or negative > rather than as above or 
below zero. 

5. The temperature stands at zero. Its first change 
is described by the expression + 6°. Its next 
change is described by + 4°. What is the tem- 
perature at the end of the second change ? 

6. If the temperature reading is 0°, and it makes 
the change — 5°, then — 3°, what is the final 
reading ? 



SECOND ILLUSTRATION: OPPOSITE NUMBERS ON A- 
DISTANCE SCALE 

EXERCISE 57 

1. An autoist starts from a certain point and goes 
east 10 miles, and then east 8 miles. How far 
and in what direction is he from the starting 
point ? 

2. If he had first gone west 10 miles, and then west 
8 miles, how far and in what direction would he 
have been from his starting point ? 

3. If he had first gone east 10 miles and then west 
8 miles, how far and in what direction would he 
have been from his starting point ? 

4. If he had first gone west 10 miles, and then east 
8 miles, how far and in what direction would he 
have been from his starting point ? 



1 68 Fundamentals of High School Mathematics 

These examples show that we must distinguish between 
opposite distances, those east of some starting point, and 
those west of the starting point. People have agreed to 
call distances east of the starting point positive and distances 
west of the starting point negative. By this means a great 
deal of time can be saved, because a positive or negative 
number tells both the direction and the distance of a point on 
the distance scale, from some beginning point. Thus, on 
the distance scale, Fig. 118, point A is completely described 
by the number — 5. % 

West A East 

I 1 1 T 1 1 I 1 1 — I 

-20 -15 -10 -5 +5 +10 +15 +20 +25 

Fig. 118. Points on a distance scale. 

This number, — 5, tells that the point A is 5 units west 
of, or to the left of, the starting point. 

5. What would be the position on this distance 
scale of a man who starts at the zero point, goes 
east 60 units, and then west 15 units ? 

6. Where would you be if you started at. zero, went 
4- 8 units, and then —8 units ? 

7. A man starts at ; at the end of the first day he 
is at + 20, and at the end of the second day 
he is at — 10. What is the total distance he 
traveled ? What number will completely de- 
scribe his position at the end of the second day? 

8. How far is it from + 9 to — 6 ? What direction 
is it ? 

Section 71. Thus, positive and negative numbers are 
used to distinguish between opposite qualities. The fore- 
going examples show that we need a brief, economical way 



The Use of Positive and Negative Numbers 169 

to denote opposite qualities of numbers. This is done by 
positive and negative numbers, or, -as we shall say from 
now on, by SIGNED NUMBERS. Thus, in referring to tem- 
perature readings, e.g. the "signed" number, -f- 10°, shows 
(1) how far and (2) in what direction the mercury stands 
from the zero point. In describing the location of a point 
on a distance scale, the " signed " number, — 6, tells how 
far and in what direction the point is from the starting or 
zero point; that is, 6 units to the left of, or to the west of, 
the zero point. 

THIRD ILLUSTRATION: OPPOSITE NUMBERS USED TO 
REPRESENT FINANCIAL SITUATIONS 

Section 72. Positive and negative numbers, or SIGNED 
NUMBERS, are used also to describe financial situations. 

It has been agreed to consider money that you " have " as 
positive and money that you "owe" as negative. Thus, 
if you owe 40 cents {i.e. — 40 cents) and have 55 cents 
(+55 cents), your real financial situation is +15 cents. 
Why? Or, if you owe 90 cents (—90 cents) and have 15 
cents ( + 75 cents), your real financial situation is — 15 cents. 

FOURTH ILLUSTRATION: OPPOSITE NUMBERS ON 
A TIME SCALE 

Signed numbers are used also to distinguish " time be- 
fore " from "time after" a given time. For example, if 
time before Christ is negative, then time after Christ is 
positive. Thus, on the time scale below, since Christ's 
birth is regarded as zero, if a man was born 10 years be- 

B.c- , , V{ L aVi f ? ft,)-r^ B . a.s+ 

•25 -20 -15 -10 S O +5 +10 +15 +20 +-25" +30 +35 

Fig. 119. Points on a time scale. 



170 Fundamentals of High School Mathematics 

fore Christ and lived 35 years, the distance between the 
points A and B would represent the period of his life. 
Why ? 

OTHER ILLUSTRATIONS OF THE USE OF SIGNED 
NUMBERS: FOR THE PUPIL TO DEVELOP 

EXERCISE 58 

1. Show how signed numbers are helpful in deal- 
ing with latitude ; with longitude. Illustrate 
each one. 

2. Show that signed numbers are a convenience 
in keeping scores in games in which you either 
make or lose a certain number of points. 

3. Can you think of any other illustrations of 
opposite numbers ? 

EXERCISE 59 
PRACTICE IN USING SIGNED NUMBERS 

1. Your teacher's financial situation is — $250. 
What does this mean ? 

2. A man's property is worth $5200 and his debts 
amount to 13300. How can positive and 
negative numbers be used to represent these 
amounts ? What number will describe his net 
financial situation ? 

3. The mercury at 8 a.m. was at — 6°. If it was 
rising 3° per hour, where was it at 9 a.m. ? at 
10 a.m. ? at 11.30 a.m. ? 

4. Show on a time scale that Caesar began to rule 
the Roman people 31 years B.C., and ruled for 
45 years. 



The Use of Positive and Negative Numbers 171 

5. What is the total number of miles traveled by a 
man who starts at zero on the distance scale if 
he is at + 6 at the end of the first day, — 2 at 
the end of the second day, and at — 8 at the 
end of the third day ? 

6. On the distance scale, where would you be if 
you started at — 4 and went east 6 miles ? 

7. If your financial condition is + 60 cents, — 15 
cents, and — 12 cents, what single number will 
accurately describe your net financial situation ? 

8. What was your final score in a game in which 
you made the following single scores : + 15, 
- 8, - 10, + 14, and + 15 ? 

9. Represent on a distance scale (horizontal) the 
point where a man would be at the end of the 
third day if he started at zero and walked + 6 
miles on Monday, — 10 on Tuesday, and — 3 
on Wednesday. 

10. Find the net financial situation of a man who is 
worth the following: (a) + $5+$8 + fl0 
-$6; (b) +6d-10d-$d+15d. 

Section 73. Absolute value of positive and negative num- 
bers. The numerical value of a positive or negative num- 
ber, without regard to its sign, is its absolute value. For 
example, the absolute value of + 6 is 6 ; of + 17 is 17 ; of 
— 9 is 9, etc. 

Section 74. We need to be able to add, subtract, multiply, 
or divide signed numbers. Now that we see clearly the 
practical ways in which positive and negative numbers are 
used we need to be able to solve problems which contain 
either kind. In all the examples which we have worked 



172 Fundamentals of High School Mathematics 

previously, only positive numbers have been used. Next, 
therefore, we must learn (1) how to combine signed num- 
bers {i.e. add them) ; (2) how to multiply them ; (3) how 
to find the difference between two signed numbers ; and 
(4) how to divide signed numbers. We will take them up 
in that order. 



I. HOW TO COMBINE SIGNED NUMBERS: FINDING 
ALGEBRAIC SUMS 

Section 75. When the numbers are arranged vertically. 

In the example : " Find the net financial situation of a man 
who is worth the following: + |5, +$8, -$10, -$6," 
we found one signed number which described the man's 
net financial situation; namely, — $3. That is, we found 
one signed number which was the result of putting several 
signed numbers together. This process is called combining 
signed numbers, or finding the algebraic sum. Thus, to 
combine +4, — 2, — 6, and 4- 3, we must find one signed 
number which is the result of putting all of these together. 
Evidently, this must be — 1. Similarly, combining, or 
finding the algebraic sum of + 5 d and — 11 d, we get 
-6d. 

In each of the following examples, find the algebraic 
sum, i.e. find one number which will describe the result of 
putting all the separate numbers together. When no sign 
is given it is regarded as positive. 







EXERCISE 60 






1. 


+ 16 


2. - 7 d 3. + 4:X 


4. 


5a 




-13 


+ 8d - Zx 




±a 




+.$4 


-4d - x 




— 6a 



The Use of Positive and Negative Numbers 173 



5. 



9. 



13. 



%y 


6. 


+ 8 


7. 


- 10 


8. 


+ 3£ 


-by 




— 5 




+ 13 




- bb 


- y 




- 6 

+ 7 




- 8 
+ 5 




-6b 

+ 2b 


±a 


10. 


-Sx 


11. 


\bab 


12. 


— 1 xy 


— 6a 




— bx 
+ 8x 




12 ab 




+ II.27 

— %xy 


+ 2|* 


14. 


- 4 


15. 


X 


16. 


%a 


-1 x 




+ 3 




2x 




-±a 


+ 2 x 




+ 11 

- 8 

+ 1 




-6x 




la 
- 9a 



Section 76. When the numbers are arranged horizon- 
tally. The numbers to be combined are almost always 
written in a horizontal line, rather than in a vertical column. 
Combining these terms is done in the same way as if 
they were written in a vertical column. For example, 
+ 6-5 + 4-9 = - 4. 

17. 6 - 8 - 5 + 11 = ? 

18. -8-9 + 11 + 6 

19. —6^+5^+9^—2^ 

20. 5 abc + 6 abc — 7 abc 

21. 4/- 3/+ 6f- 9/ 

22. 4- 8/- 9t - 6t + 12 t 

23. How have you found the algebraic sum of 
these numbers ? 

24. ±x—dx—2x + x — 10x 

25. — 3 ab + 5 ab — 8 ab + ab 



1 74 Fundamentals of High School Mathematics 



26. 2t+5t-9t—t 

27. a 2 + la 2 -1 a 2 -9 a 2 + 2 a 2 

28. — ^ 4- 12/ — lby+y — Sy 

29. 5x — x—2x — 3x — 4:x — 10x 

so. 8-5-9-2-7 + 10 

31. j- 3 + 6^ 3 +2j 3 -5j 3 -12^3 

32. #&■ — 7 #&: — 9 #&: — 3 abc 

Section 77. Terms are either LIKE or UNLIKE: How 
to distinguish TERMS. Any algebraic expression, such as 
ax + b or x 2 + 2 xy + 5, is made up of one or more num- 
bers separated from each other by 4- signs or by — signs. 
These numbers thus separated from each other are called 
terms. Thus, in ax 4- b there are two terms ; ax is one, 
b is the other ; while in x 2 4- 2 xy — 5 there are three terms, 
x 2 , 2xy, and 5. Note carefully that a "term" includes 
everything between 4- or — signs. 

In many algebraic expressions these terms are all like 
terms, and, as we learned in the previous section, can 
be combined or put together into one number or term 
which we called their algebraic sum. Thus, 4 d, 4- 5 d, 

— Sd, 4-. 3d? are similar or like terms, and their algebraic 
sum is + 4 d. It is important to understand that because 
each letter in the expression represents the same thing 
these are like terms. In many cases, however, the terms 
of an algebraic expression are not all like terms. 

For example, consider : 4 boys, 4- 5 girls, 4- 8 boys, 

— 2 girls, or, using the initial letters of the words, 4 b 4- 
5^-4-8^ — 2^-. Evidently, these are unlike terms and 
cannot be combined into one number. However, the like 
terms in the expression can be combined ; that is, the 4- 4 b 
and 4- 8#, giving 12 b, and the + 5g and —2g, giving 



The Use of Positive and Negative Numbers ' 175 

+ 3^-. Thus the expression 4 b + bg + 8 b — 2g can 
be simplified or expressed more briefly by combining like 
terms, giving 12 b + 3^-. From this illustration we see 
that the like terms of any algebraic expression can be 
combined, giving a simpler, briefer expression than the 
original one. 

EXERCISE 61 

FURTHER PRACTICE IN COMBINING SIGNED NUMBERS: NUMBERS HAVING 
LIKE OR UNLIKE TERMS 

Write in the simplest or briefest form each of the 
following expressions : 

1. 2a + 3a - 6a + 4a 

2. 5 ft. + 6 in. - 2 ft. - 4 in. 

3. 7 yr. + 3 mo. — 2 yr. — 1 mo. 

4. 4c b + 5c- Sb -1c + b 

5. 6a 2 +3a 2 -7a 2 + 4a 2 

6. - 2x* -5x*- 8^ + 2x* 

7. ^ + 5 + 4^ + 3 

8. 3xy + 5 ab -7 xy -11 ab 

9. 2 b* - 7 b* + 5 6 3 - b 3 

10. 2r + 8r+3r- 10 r 

11. 4a 2 b-{-5a 2 b-8a 2 b-Sa 2 b 

12. -6 + 4-8+6-9 + 2 

13. 5^+3 — 8^-4 + 4^ + 1 

14. tf 2 £ + 4tf 2 £-6rt 2 £ 

15.' ^7 + 3 — 8 xy — 9 + 2 .27 + 7 

16. / + 2^-8/ + 4^ + 6/ + 5? 

17. 3;r+ 57 — 7;r — 87 — ;r +7 



176 Fundamentals of High School Mathematics 



5 ab + 6 xy — 6 ab — 7 xy + 2 ab + 2xy 
- 3 b 2 + ±a 2 - 1 b 2 - 9 a 2 + 12 b 2 - 3 a 2 
%f-t- 11 f + 5 / + 3 f - 9 / 
4 + 6*-- 9 -11*-+ 2 + 7*- 1 

^ 3 + £ 3 - 2 



18. 
19. 
20. 
21. 
22. 

23. 2/ — # — t+5x+l t — 10 

24. 5j/ 2 -4+11j/ 2 + 7 + 6jj/ 2 -3 
"-. 7 «^ 2 - 9 « 2 ^ - 11 abc 2 + 11 a 2 bc + ^^ 2 



3^3 + & ^3_ 2 



25 



SUMMARY OF IMPORTANT PRINCIPLES CONCERNING THE 
COMBINING OF SIGNED- NUMBERS 

Section 78. You have now worked many examples in 
finding algebraic sums. From your experience with such 
examples, complete these three sentences which tell how to 
combine signed numbers : 

1. To find the algebraic sum of two positive numbers, 

I the absolute values of the numbers, and 

give to the result a I sign. 

2. To find the algebraic sum of two negative numbers, 

? the absolute values of the numbers, and 

give to the result a I sign. 

3. To find the algebraic sum of two numbers which 
have unlike signs, find the ? of their abso- 
lute values, and give to the result the sign of 
the number which has the I absolute value. 



PRACTICE EXERCISE B: COLLECTING TERMS 

Practice on this exercise until you can reach the standard, 
10 examples right in 3 minutes. Record the number which 
you try and the number which you do right. Compare your 
record each time with the standard. 



The Use of Positive and Negative Numbers 177 



!. 2.r 2 +3-5,r 2 -7+4;r 2 -6 




2. -2b + c + 5b-±c-2c-6b.. 




3 . $ a s_2b+7 a* + 3b-15a 3 -b . 




4. 5-3^ 2 + 4-^ 2 + 2^-8 




5. ac 4- 6 - 3 ac - 13 - 2 ac 4- 5 .... 




6. 2y-5*-8y-4* + 6y+9;r. 




7> 2£ 2 -3£-5£ 2 -7£4-6£ 2 4-8£ . 




8. 4/ - 9 - 7/ + 6 - 8/ - 13 




9t 7_8^ 2 -ll + 5^ + 4 + 3 5 2 




10. — ax — 5 «.r + b — 2 £ + 4 tf.r 4- 5 b 




11. 3j-5/ 5 + 8i--14j-+6^ 5 -ll/ 5 




12. -12-8^-4-9 w + 16 4- 17 «/ 




13. _ 3 e s + 4 - e* + 1 + 5 ^ 3 - 7 




14. 2,r+ 3j— 11 ;r — 13y + 5x — 7j/ . 





Section 79. Equations solved by addition of signed num- 
bers. What you have just learned about signed numbers 
will now be used in solving equations. For instance, in 
solving the equation 

n-10 = -6 

it is necessary to add 4- 10 to each side of the equation. 
This gives the equation 

;/ = + 4 
Again, in solving a difficult equation such as, 
5y - 24 = - 2 y - 3 

it is necessary to add 4- 24 to each side. (Why ?) This 
gives the equation 

5y = - 2y + 21 



178 Fundamentals of High School Mathematics 

Then, - 2y must be added to each side. (Why ?) This 
gives the equation 

ly = 21. 



EXERCISE 62. Oral Work 



1. *-8= + 10 

2. y- 2 = + 2 

3. 5=*-2 

4. /-7=-2 

5. /_6 = -4 

6. *-8 = -10 

7. *-10 = -12 

8. 7 -6=-8 

9. / — 4 = — 4 

10. t-7 = -10 

11. £ - 7 = + 8 

12. 2*= 15-.* 

13. 3j/ = 12-j 

14. 2£ = 20-2£ 

15. £-10 = -2 



16. 5y = 16-3y 

17. 4^ = -^ + 15 

18. £ = -2£ + 12 

19. y=-y + \Z ) 

20. 2^ = 10-3^: 

21. 10+*= -10 

22. 8+2y = + 12y 

23. 5j/— 2 = 18— y 

24. 8/- 5 = 35 -2/ 

25. 7r-8 = -r+24 

26. 4£-l = -2£ + 17 

27. 12/ -100 = -13/ + 25 

28. 4*- 50 = 70-* 

29. 7*-5*-8*4-14* = 30 

30. -5y — 4:y—y + 12y = 18 



II. HOW TO MULTIPLY SIGNED NUMBERS 

Section 80. The four ways to multiply signed numbers. 

In arithmetic it was found that multiplication shortened 
the work of addition. For example, in adding 3 + 3 + 3 + 
3 + 3 + 3 + 3, the result is found most easily by multiply- 
ing 3 by 7, because 3 is taken 7 times. So, in algebra, it 
is equally desirable to multiply one signed number by 
another. 



The Use of Positive and Negative Numbers 179 

There are four different ways in which we may have to 
multiply signed numbers. These are : 

(1) plus times plus, as in the example + 4 times 
+ 2 = ? 

(2) plus times minus, as in the example -f- 4 times 
— 2 = ? 

(3) minus times plus, as in the example — 4 times 
+ 2 = ? 

(4) minus times minus, as in the example — 4 times 
-2=? 

By considering the following problems we can tell what 
meaning must be given to the multiplication of signed 
numbers. 

A. ILLUSTRATIVE QUESTIONS BASED UPON THE SAVING 
AND WASTING OF MONEY 

EXERCISE 63 

1. If you save $5 a month (+ $ 5), how much better 
off will you be 6 months from now (+ 6) ? 
Evidently you will be 1 30 better off ( + $30). 
Thus, 4- 5 times + 6 = + 30. 

2. If you have been saving $5 a month (+ $5), how 
much better off were you 6 months ago (— 6)? 
Evidently you were $30 worse off (— $30) than 
you are now. Thus, -f 5 times — 6 = — 30. 

3. If you are wasting $5 a month (— $5), how much 
better off will you be in 6 months from now 
(+6)? 

Evidently you will be $30 worse off (-$30). 
Thus, - 5 times + 6 = - 30. 

4. If you have been wasting $5 a month ( — $5), 



180 Fundamentals of High School Mathematics 

how much better off were you 6 months ago 

(-6)? 

Evidently you were 130 better off (+830). 

Thus, — 5 times — 6 = + 30. 

Summarizing : These problems based upon saving and 
wasting money have led to the following illustrative state- 
ments : 

1. 4- 5 times + 6 = + 30. 

2. 4- 5 times - 6 = - 30. 
3.-5 times + 6 = - 30. 
4.-5 times — 6 = + 30. 

B. ILLUSTRATIVE QUESTIONS BASED UPON 
THERMOMETER READINGS 

EXERCISE 64 

1. If the mercury is now at zero and is rising 2° 
per hour (+ 2), where will it be 4 hours from 
now ( + 4)? 

Evidently it will be 8° above zero (+ 8). Thus, 
+ 2 times +4- + 8. 

2. If the mercury has been rising 2° per hour (+ 2°) 
and is now at zero, where was it 4 hours ago 
(-4)? 

Evidently it was 8° below zero ( — 8°). Thus, 
+ 2 times - 4 = - 8. 

3. If the mercury is now at zero and is falling 2° 
per hour ( — 2°), where will it be 4 hours from 
now ( 4- 4) ? 

Evidently it will be 8° below zero (- 8°). Thus, 
- 2 times + 4 = - 8. 



The Use of Positive and Negative Numbers 181 

4. If the mercury is now at zero and has been fall- 
ing 2° per hour (— 2°), where was it 4 hours ago 
(-4)? 

Evidently it was 8° above zero ( + 8°). Thus, 
- 2 times - 4 = + 8. 

Summarizing : these problems based upon the ther- 
mometer have led to the following illustrative statements : 

1. +2 times + 4 = + 8. 

2. + 2 times - 4 = - 8. 
3.-2 times + 4 = - 8. 
4.-2 times - 4 = + 8. 

A careful study of these illustrations will enable you to 
complete the following statements concerning multiplica- 
tion of signed numbers : 

1. A positive number multiplied by a positive num- 
ber gives as a product a I . number. 

2. A positive number multiplied by a negative 
number gives as a product a 1 number. 

3. A negative number multiplied by a positive 
number gives as a product a 1 — number. 

4. A negative number multiplied by a negative 
number gives as a product a I number. 

5 . The product of two numbers which have like signs is ? 

6. The product of two numbers which have unlike signs is 



The last two statements are generally used as rules for 
multiplication. These rules or general statements are 
based upon the previous illustrations. You should refer 
to them when in doubt about how to multiply any two 
signed numbers. 



1 82 Fundamentals of High School Mathematics 



n, 



PARENTHESES ARE USED TO INDICATE 
MULTIPLICATION 



Section 81. Multiplication of two or more numbers is 
often indicated by placing the numbers within parentheses. 
Thus, " + 4 times -35" is often written " ( + 4)(- 35)." 
It is important to note that no sign or symbol is placed 
between the parentheses when multiplication is indicated. 





EXERCISE 65 






PRACTICE IN MULTIPLYING SIGNED NUMBERS 


1. ( 


' + 3)(+5) 




19. ( 


;3)(4)(5)(2) 


2. < 


:+«)(-2) 




20. ( 


+ 2)(+$5) 


3. ( 


;+io)(-2i) 




21. ( 


>3)(+7Z>) 


4. { 


:+6x-9) 




22. 


; + 6)(+5ft.) 


5. I 


;-2)(-5) 




23. ( 


-8)(+6 7 ) 


6. < 


:+8)(-i) 




24. ( 


+ f)(18a) 


7. I 


:+i2)(+6) 




25. 


:-12)(+i) 


8. 


;+5 )(+4)(-2) 




26. ( 


IX -18) 


9. i 


^3.)(-6)(+2) 




27. ( 


27)(-f) 


10. ( 


;-4)(-10)(-3) 




28. ( 


+ IX-25) 


11. ( 


:+!X-i) 




29. ( 


-32)(-f) 


12. < 


;-»(.+*) 




30. ( 


-IX + 240 


13. ( 


_ 7 )(-6)(+2) 




31. ( 


-IX+I) 


14. | 


;-2)(-2)(-2) 




32. ( 


-*)(-«) 


15. ( 


;-3)(-3)(-3) 




33. ( 


-2)(-2)( + 


16. 


: + l)( + l)( + l)( + l) 


34. ( 


+ i)(-if) 


17. ( 


;-2)(-2) 




35. ( 


iX-io) 


18. ( 


;-2)(-2)(-2)(- 


2) 


36. 


-5-8 



The Use of Positive and Negative Numbers 183 

37. -7-21 41. (2)(-3)( + 4)(-5) 

38. (-6X-1) 42. (6)(I)(|) 

39. 2- 2- (-3) 43. (10)(5)(-Jo) 

40. (_1)(-1)(-1)(-1) 44. (1)(-A) 

II b. HOW TO USE EXPONENTS IN MULTIPLICATION 

Section 82. Suppose we had to find the product of 3x 2 
and 5;r 4 . It is important to keep in mind the meaning of 
exponents. 3 x 2 means 3 • x • x and 5 x^ means 5 • x • x • x • x. 
Hence, 3x 2 times Bx 4 or (3x 2 )(5x A ) means 3 • x • x • 5 • x • x 
• x - x, or 15 x Q . By the same reasoning, the product of 
+ 6x 5 and — 7 ^r 4 is -42;tr 9 . 

IMPORTANT PRINCIPLE OF USING EXPONENTS IN 
MULTIPLICATION 

From such examples as these we can state a very im- 
portant principle : 

The exponent of any letter in the product is equal to the 
sum of the exponents of that letter in the separate factors. 

EXERCISE 66 

PRACTICE IN USING EXPONENTS IN MULTIPLYING SIGNED NUMBERS 

(ORAL) 

1. (5a 2 )(6tf 4 ) s. 2ab 2 -Zab 

2 . (+7£)(_9£5) 10. a 2 b-ab 2 .aW 

3. ( + 8)(2j/ 3 ) 11. x.Zx* 

4. (6ab)(2a 2 ) 12. -6a -Za 

5. (+%abc)(bab) 13. -2^-3/ 

6. 4^ 3 .5^ 2 .2^ 5 14. (~5b)(-2b 2 ) 

7. / • 5 y 15. ( - 6 ^ 2 )( - 7 *^) 

8. i^-lO* 7 16. (-8^ 2 )(i^) 



:84 Fundamentals of High School Mathematics 



17. (+10j/V)-(2^ 2 ) 

18. 16a a A.(£*£ 8 ) 

19. a > b • b > a > a 

20. ^r 4 • 2x 

21. _>' • 5 J/ 3 

22. b 2 c • &: 

23. 5 • ;tr 2 

24. 10 j/ 2 • Jq-jJ/ .J 3 

25. X 2 • 3 * • X* 

26. -2j/.j/ 2 

27. (-^)(_5)(^ 2 ) 

Write a rule for the use of exponents in multiplication. 



28. ("/)( + /)(-/) 

29. (J **)(- 6 0(f) 

30. (-!*)(- i* 5 ) 

31. (--^)(-l)C-l) 

32. (^)(~«%) 

33. (+8y*r)(-J;ry) 

34. (-10*)(^>) 

35. (_^a)(_^«)(_y) 

36. (-/)(+/)(-/) 

37. ("!-/)(- **)0)' 



REVIEW EXERCISE 67 

Perform the indicated operations in the following 
examples : 

1. 5a 2 + 2b-8a 2 -b + a 2 . 

2. (4^ 3 )(2^)(-3^ 2 )(-^). 

3. Represent, by a drawing, two squares, each 
side of the first being 2 a, and each side of the 
second 4 a. 

(a) What is the perimeter of each square ? 
What is the ratios of their perimeters ? 

(^) What is the area of each square? What 
is the ratio of their areas ? 

4. A final examination contained the following 
question : Give both the algebraic sum, and the 
product, of the following expressions : 



The Use of Positive and Negative Numbers 185 



(a) -8, +2, -1, +10, -1 
00 4.r, 5;r 2 , Zx, -Sx 2 . 
(0 f, 2y, - 5y, - 7/. 

Why do you think this was a difficult question ? 

5. What is the shortest way to write the sum of 
five ;r 2 's ? the product of five x 2 's ? 

6. Evaluate a 2, when a = — 1 ; when a — — 2 ; 
when a = — 5. 

7. Determine the numerical value of a 2 b — ab 2 
when a = 5 and b = 2. 

8. .r 2 + ^ + 2^2 + 2^r 3 + 3^ 2 . 

9. ;r 2 .#*. 2^ 2 - 2x*.Sx*. 

10. What algebraic expression will represent the 
area of Fig. 120 ? Evaluate the expression when 
x= 5 and y = 4.5. 









y 

X 



2X 

Fig. 120 




11. Find the area of all the faces or sides of the 
rectangular box, Fig. 121. What expression 
represents the volume ? What would the 
volume be if each dimension were 4 y ? 

12. Add: 4^-3j/4-2, 5x + 67 -8, -7x + 3y 
-9. 



1 86 Fundamentals of High School Mathematics 



III. HOW TO FIND THE DIFFERENCE BETWEEN SIGNED 
NUMBERS: SUBTRACTION 

Section 83. How " differences " are found in practical 
work. Clerks in stores have a method of making change 
or of finding the difference between two numbers that is 
very helpful in finding the difference between two signed 
numbers. For example, if a customer gives the clerk 50 
cents in payment for a 27-cent purchase, the clerk begins 
at 27 and counts out enough money to make 50 cents. If 
we use the same terms as were used in arithmetic, — 
namely, the subtrahend, minuend, and difference, — then 
we say, " The clerk begins at the subtrahend, 27 cents, 
and counts to the minuend, 50 cents." 

First illustrative example. To illustrate this method of 
finding the difference between two signed numbers, let us 
consider this problem : 

On a certain day the mercury 
stands at — 4° in Chicago and 
at + 13° in St. Louis. How- 
much warmer is it in St. Louis, 
or what is the difference between 
+ 13° and - 4° ? Naturally, we 
do the same thing the clerk does, 
begin at the subtrahend and count to 
the minuend, i.e. we count from 
- 4° to + 13°, giving us + 17°. 
The difference is called positive 
because we counted upward. If 
we counted downward, the dif- 
ference would be called negative. 
This example is written as fol- 
lows: 

+ 13° minuend 

— 4° subtrahend 

+ 17° difference 



-4+ 



no -j 


hio 


loo -| 


|100 


•90 4 


h90 


■80 J 


\Q0 


•70 -4 


H70 


-60 -| 


Uo 


-50 -| 


r° 


-30 -| 


Uo 


-30 -| 


H30 


-20 H 


ho 


"° I 


r° 


io -| 


hio 


•20 A 


ko 



a 



Chicago St. Louis 
Fig. 122 



The Use of Positive and Negative Numbers 187 



Second illustrative example. Subtract 
+ 10 from — 5 by referring to the number scale. 
This means to find the distance from the sub- 
trahend to the minuend or from +10 to — 5. 
The distance from 10 above to 5 below is clearly 
15 ; and since the direction is downward, the 
difference is — 15. This example is written : 

— 5 minuend 
+ 10 subtrahend 



— 15 difference 



+ 35 
+ 30 
+ 25 
+ 20 
+ 15 
+ lO 

B+ 5 
4i± O 

- 5 
-lO 

H-15 
-20 

Fig. 123 



These illustrations are given merely to show that the 
difference between two signed numbers can always be 
found by counting on a number scale from the subtrahend 
to the minuend. The difference will be positive or nega- 
tive, depending upon whether the direction of counting is 
tip ward or downward. 

EXERCISE 68 



PRACTICE IN FINDING THE DIFFERENCE BETWEEN TWO SIGNED 
NUMBERS: SUBTRACTION 



1. 



3. 



4. 



+ 6 - 


-4 4-8 4-3 


+ 5 


-T +13 +9 


_2 4-5 4-2 4-10 


-8 


+ 1 + 


4 +14 


+ 14 


- 6° + 7 d 


+ 10 ft. 


— 4 in. 


+ Sx 


-18 


4-9° -2d 


- 6 ft. 


-7 in. 


-10* 


-3a 


+ ±b -oe 


10 x 2 


-3**j? 


+ 10 abc 


-11a 


-lb + §c 
4- bxf 


-2x 2 


+ 5 x s y 
Sx 


— 4 abc 


-2fi 


5 a 


-2 be 


4- 3 fi 


4-11 xf 


-2a 


X 


be 



1 88 Fundamentals of High School Mathematics 

5. From —la take + 5 a 

6. Take - 13 b 2 from + 2 b 2 

7. From x take 7 ;r 14. Take — 6 from — 4 

8. Take — be from 2 fo 15. From + 2 take — 15 

9. (6) -(-2) 16. From + 12 j/ 2 take -8j/ 2 

10. ( + 4) - ( + 4) 17. Diminish + 6 by - 4 

11. (-2)-(+8) 18. (+2) + (-8)-(-6) 

12. (_3j0-(+2 7 ) 19. (-2)-(+6)-(-10) 

13. (8^ 2 )-(10^ 2 ) 

20. Find the value of a — b, if a — 10, b = — 5. 

Section 84. Subtraction without the use of a scale. In 

arithmetic it is often stated that subtraction is the process 
of finding what number must be added to one number, the 
subtrahend, to produce another number, the minuend. 
For instance, in subtracting 12 from 20, you are to think 
what number must be "put with" or "added to" 12 to 
give 20. Obviously 8 is the number. Let us apply the 
same thinking to signed numbers. 

Illustration. Find the difference between + 12 and — 4. 
Think what must be " put with " or " added to " -f 12 
-4 to give + 12. Clearly it is + 16 — 4 

+ 16 

Thus you see another method of subtracting signed num- 
bers, i.e. you find out what number must be "put with " or 
"added to" the subtrahend to produce the minuend. 

EXERCISE 69 

Solve these examples by thinking as you did in the 
explanation given above. 



The Use of Positive and Negative Numbers 189 



7. + o 
+ 11 



1. 


+ 10 4. + 2 




-2 -18 


2. 


+ 8 5. +12 




-6 +15 


3. 


+ 12 6. + 4 




-10 +9 


16. 


— 7 xy 




-12xy 


17. 


+ 2x + Zy 




— ox— Sy 


18. 


— 3 / + 5 w 




+ 8t-2w 


19. 


2x — y 




— 7 x+ 5y 



8. + 


2 


+ 


8 


9. — 


8 


— 


8 



10. 


-12 




— 7 


11. 


-25/ 




-18/ 


12. 


-11 X 




— 7 x 



13. 


- 3 




- 9 


14. 


- 5y 




-lly 


15. 


- Sx 2 




-12;r 2 



20. (12) -(-3) 

21. (-15) -(+12) 

22 . (_i4)-(+14) 

23. 5x—(—4:x) 

24. l-(-6) 

25. x — (5x) 

26. \y — (— 3jj/) 



Section 85. Solving equations which necessitate the sub- 
traction of signed numbers. In Chapter II you solved 
equations by subtracting certain numbers from each side. 
For example, the equation Sx+ 10 = 22 is solved by sub- 
tracting 10 from each side. In such examples the subtrac- 
tion did not require the use of signed numbers. However, 
if we wish to solve an equation such as 

3^r+10 = -8, 

we must subtract + 10 from each side of the equation. 
That requires us to subtract signed numbers ; that is, we 
subtract + 10 from — 8, which gives us — 18. Then we 
have the equation 

3;r=-18 



or 
x=-6 



190 Fundamentals of High School Mathematics 



EXERCISE 70 
PRACTICE IN SOLVING EQUATIONS WHICH INVOLVE SUBTRACTION 



1. ^ + 8 = 10 

2. 2j/ + 3 = ll 

3. 5^ + 1 = 21 

4. 6a + 2 = 20 

5. ^-+12 = 10 

6. 7 + 6=4 

7. £ + 11 = 7 

8. ^ + 2 = 2 

9. 7 + 5 = - 1 

10. £ + 4 = -8 

11. ^ + 10 = -4 

12. j/+15 = -3 

13. £ + 4 = -4 

14. /+9 = -9 

15. ^ + 10 = 10 



16. 11 + ^ = 8 

17. 27 + 10 = 8 

18. 9 b + 12 = 3 

19. 4 + 6 *=2 

20. 8 +7 = — 4 

21. 6 + 4 ^ = — 6 

22. 10+/ = -2 

23. 12 +57= -13 

24. 16 + 8x = -16 

25. 2/+ 20 = -4 

26. 6£+4 = 2£-8 

27. 10 + %p = - 4 +p 

28. 2*- 10 = 12 

29. 37-8 = -20 

30. 10^-2= £ + 16 



IV. HOW TO DIVIDE SIGNED NUMBERS 

Section 86. Division is the opposite of multiplication. 

You will have little or no difficulty in the division of 
signed numbers if you understand that division is just 
the opposite of multiplication. For example, if 4 x 2 = 8, 
then I = 4. In this case 8 is the dividend, 2 is the divisor, 
and 4 is the quotient. In signed numbers, as well as in 
arithmetic, the dividend equals the quotient times the 
divisor. 

+ 8 -f- - 2 = - 4 ; or ±| = - 4, because ( - 2)( - 4) = + 8. 



The Use of Positive and Negative Numbers 191 



- _ 2 = + 4 ; or 



+ 4, because ( - 2)(+ 4)= - 8. 



+ 8 



+ 8-^ + 2 = +4; or -^ = + 4, because ( + 2)( + 4)= + 8. 

+ ^ 

_8- + 2 = -4;or^! = -4, because ( + 2)(- 4)= - 8. 
4- 2 



EXERCISE 71 
PRACTICE IN FINDING THE QUOTIENTS OF SIGNED NUMBERS 

Find the quotient in each of the following : 

+ 10 +18. -16. -30. -14. + 6. -8. +16 



+ 2' 

+ 15 d, 
-3 ' 



' +4 ' -10' +7 ' +2' -1' 

18 ft , +16 mo. . + 25* . -\0a 

+ 3 ' -8 ' -5 ' +2 ' 



3. (21)+(-8); (-36K(-9); (-54)+(+ 27); 

(_96)^(+12); (_21)-(-7); (_60)^( + 12). 

A 10„r 3 2iy 18 b b 12^ 16/ 3 „ 

4 - T^ ; ly ; ^ ; T^ ; 167* Howcan y° u 



prove each of these 



5. 



+ 20 j/ 2 . -27 ;g* . -34£V. -50^ . -72yW> 
__4^ ' _9.r 4 ' -17£ 2 ' + 25j/' .- 6j/w 5 ' 



€. What is a good way to check or prove your 
work in division ? 



EXERCISE 72 
PRACTICE IN DIVISION WHICH INVOLVES SIGNED NUMBERS 

l. Divide each of the following by 2 a : 4 a, 6 a 2 , 
-8 a, -10 a 2 , a, -12a*, +12 a 5 , - 6 a 2 x, 
- 11 a*y. 



192 Fundamentals of High School Mathematics 

2. Divide each of the following by — by : 10j/, 

— by 2 , — 15 j 3 , y 2 , —y s , 10 xy, — 2bx 2 y s } — xy. 

3. If — 3j/=24, what does -\-y or y equal? If 

— bx = 30, what does + x, or x, equal ? 

4. Solve each of the following equations : 



0) -2* = 12 

(b) -by = 20 

(c) 4 7 = 12 

\d) -6 £ = -18 
(*) -9^ = 90 

(/) -^ = 10 

(g) -2x=-U 



(//) -5.r+2 = -8 

(1) _ 4 3 - 3 = - 19 

O) 7 - 87 = - 49 

(k) 2-c=-b 

(/) 5 + 3j/=-4 

(*») 14 - 5 £ = ' - 6 

(«) 5— 7 = — 4 



EXERCISE 73 
COMPLETING STATEMENTS ABOUT DIVISION 

1. A positive number divided by a positive number 
gives as a quotient a ? number. 

2. A negative number divided by a positive number 
gives as a quotient a ? number. 

3. The quotient of two numbers having like signs 

is * . 

4. The quotient of two numbers having unlike signs 

is t . 

5. The exponent of any letter in the quotient is 
equal to the exponent in dividend ? the 
exponent in the divisor. 



The Use of Positive and Negative Numbers 193 

EXERCISE 74 

A REVIEW OF ADDITION, MULTIPLICATION, SUBTRACTION, AND 
DIVISION OF SIGNED NUMBERS 

This is a very important exercise. 

1. From the sum of 2 a and — 5 a take the differ- 
ence between — 3 a and + 8 a. 

2. Add the product of — 3 and + 5 to the quotient 
of - 18 and - 2. 

3. Take the sum of — 7 b and + 4 b from the dif- 
ference between — 6 b and + 11 b. 

4. To the quotient of — 21 and + 3 add the 
product of — 6 and -f 7. 

5. From the sum of 7 / and — 10 t take the differ- 
ence between — 4 t and -f 11 1. 

6. Add, the product of — 6 and + 9 to the quotient 
of - 28 and - 4. 

7. Take the sum of — 9 x and + 3 x from the dif- 
ference between — 5x and + 13 x. 

8. To the product of — 7 and -f 11 add the quo- 
tient of - 33 and + 3. 

9. From the sum of 8 c and — 14 c take the differ- 
ence between — 5 c and +16*:. 

10. Add the product of — 12 and + 5 to the quo- 
tient of - 32 and - 4. 

11. Take the sum of — Sj/ and 3jj/ from the differ- 
ence between — 7 y and + 12 y. 

12. To the product of — 8 and + 9 add the quotient 
of - 36 and + 6. 

13. From the sum of 12 b and — 16 b take the dif- 
ference between —lb and + 8 b. 



194 Fundamentals of High School Mathematics 

14. Add the product of — 8 and + 9 to the quotient 
of — 40 and — 5. 

15. Take the sum of — 11 / and 7 t from the differ- 
ence between — 9 / and + 10 t. 

16. To the product of — 6 and + 13 add the quo- 
tient of — 42 and + 7. 

SUMMARY OF CHAPTER IX 

This chapter shows the following important points : 

1. The need for skill in using signed numbers to 
represent opposite qualities, such as thermometer 
readings, distances, etc. 

2. That like terms represent the same thing in an 
expression and can be combined ; unlike terms 
represent different things and cannot be combined 
in one number. 

3. To find the algebraic sum of two numbers which 
have unlike signs, find the difference of their 
absolute values, and give to the result the sign 
of the number which has the greater absolute 
value. 

4. You reviewed the principle that whatever is done 
to one side of an equation must also be done to 
the other side. For example, in your practice 
with equations you added to, subtracted from, or 
divided by the same signed number on each side 
of the equation. 

5. The product of two numbers which have like signs 
is positive; the product of two numbers which 
have unlike signs is negative. 



The Use of Positive and Negative Numbers 195 

6. The exponent of any letter in a product is equal 
to the sum of the exponents of that letter in the 
separate factors. 

7. The quotient of two numbers which have like signs 
is positive ; the quotient of two numbers which 
have unlike signs is negative. 

REVIEW EXERCISE 75 

1. The formula h = 25 + J (G — 4) is used to deter- 
mine the proper height of the chalk trough in a 
schoolroom. If h stands for the height in 
inches, and g stands for the number of the 
grade, find the height for Grade VIII; that is, 
when g = 8. What is the proper height for a 
third-grade room ? 

2. Evaluate the expression ab 2 + a 2 fr if a = 2 and 
= -3. 

3. Show that the sum of any two numbers having 
unlike signs, but the same absolute value, is zero. 
Give some illustrations. 

4. In a class of 25 pupils, 2 were conditioned and 
6 failed. Express the ratio of the number of 
pupils that succeeded to the total number in the 
class. What percentage is this ? 

5. The ratio of y + 1 to 9 is equal to the ratio of 
y + 5 to 15. Find y. 

The number of posts required for a fence is 84 
when they are placed 18 feet apart. How many 
would be needed if they were placed 12 feet apart ? 

6. If I am now x years old, what does the follow- 
ing expression tell about my age : 2 x + 5 = 55 ? 



CHAPTER X 

THE FURTHER USE OF THE SIMPLE EQUATION 

A. HOW TO SOLVE SIMPLE EQUATIONS WHICH CONTAIN 
NEGATIVE NUMBERS AND PARENTHESES 

Section 87. What we have already learned about the 
equation. Since the equation is a very important part 
of mathematics, we must be able to solve quickly and 
accurately equations of any kind. Thus far we have 
learned two very important facts about equations : 

1. That if we do anything to one side of an equation, 
we must do the same thing to the other side. 

2. That an equation is solved when a value of the un- 
known is found which satisfies the equation ; that is, one 
which makes the numerical value of one side equal to the 
numerical value of the other side, when the value is sub- 
stituted for the unknown. 

Furthermore, you have learned : (1) how to solve simple 
equations of the type 

6 b + 3 = 45, 
or, c -+- 5 c — 20 -f c, etc. ; 

(2) how to get rid of fractions in an equation, e.g. of the 
type \x+\x — 1 = 3; 

(3) how to solve word problems, first by translating them 
into equations and second by solving the equations. These 
methods, which you have now mastered, are important first 
steps in the more important problem of learning how to 
solve equations of any kind. 

I. SOLVING EQUATIONS WHICH CONTAIN NEGATIVE 
NUMBERS 

Section 88. There are just two more steps that we must 
learn in using equations. First, we must be able to solve 



The Further Use of the Simple Equation 197 

equations which contain negative numbers ; second, we 
must be able to solve equations which contain parentheses. 

Negative numbers occur very commonly in equations. 
The following examples illustrate this fact. 

EXERCISE 76 

Write as equations, and solve each of the following 
examples : 

1. What number multiplied by 7 equals — 28 ? 

2. What number multiplied by —5 equals 20? 

3. If a certain number be added to 13, the result 
is 8. Find the number. 

4. A certain number increased by 10 equals — 5. 
Find the number. 

5. If 7 be subtracted from a certain number, the 
result is — 3. What is the number ? 

6. If negative four times a certain number gives 
22, what is the number ? 

These examples show how negative numbers occur in 
equations. Throughout the remainder of the work, equa- 
tions which are satisfied by negative numbers- will occur 
very commonly. The next exercise contains many exam- 
ples of this kind. 

EXERCISE 77 
SOLUTION OF EASY EQUATIONS WHICH CONTAIN NEGATIVE NUMBERS 

Solve each of the following equations. You should be 
able to tell exactly what must be done to each side of the 
equation. 

1. * + 5 = 3 3. £ + 7 = 2 5. 2^ + 16 = 2 

2. 2^ =-16 4. -3a = 15 6. -4 y = 12 



198 Fundamentals of High School Mathematics 

7. 10.y + 2 = - 18 a 2 £-1 = 9 

9. Three times a certain number, increased by 10, 
gives 6. What is the number ? 

10. If twice a certain number is added to 16, the 
result equals the number increased by 6. Find 
the number. 

11. ?£+5=4* + !* + 2. 12. 12 -2*= 8. 

6 4 6 

13. The sum of two thirds of a certain number and 
three fourths of the same number is — 17. 
Find the number. 

A new kind of equation 

14. -%x- 12 = 5^-40. 

The equations which you have just solved are of the 
kind in which you can easily see what to do to each side. 
With examples like 14, however, in which both knowns 
and unknowns occur on each side and which include nega- 
tive numbers on one or both sides, we need special and 
systematic practice. 

Section 89. We need to get " knowns" on one side and 
"unknowns" on the other. Just as clerks in stores always 
place the known weights on one scale pan and the un- 
known weights on the other scale pan, so we, in solving 
equations, always get the known numbers, or terms, on one 
side of the equation, and the unknown terms on the other side. 

Usually we get all the unknown terms on the left side, 
and all the known terms on the right side. Thus, in the 
equation above, 

-2^-12 = 5^-40, 

we do not want — 12 on the left side. Therefore we get 
rid of the known on the left side by adding + 12 to each 
side, giving the equation —2^=5^—28. We also do 



The Further Use of the Simple Equation 199 

not want the 5x on the right side. Therefore we subtract 
5 x from each side, giving the equation — 7 x = — 28. Divid- 
ing each side of this equation by — 7, we find that x = 4:. 
Section 90. Equations should be solved in a systematic 
order. In learning to solve equations which require several 
steps, pupils make many mistakes because their work is not 
arranged in a set order. For the present, therefore, you 
will find it very important to use a form like the following : 

Illustrative example. Solve the equation 
- 2 x - 12 = 5 x - 40. 

(1) Adding + 12 to each side gives 

- 2 x = 5 x - 28. 

(2) Subtracting 5 x from each side gives 

-7x=-28. 

(3) Dividing each side by — 7 gives 

x = 4. 

(4) Check : Substituting 4 for x gives 

- 8 - 12 = 20 - 40. 
-20 =-20. 



EXERCISE 78 



Solve each of the following examples, writing out each 
step exactly as in the solution of the illustrative example : 



1. - 3*- 8 = 8*- 30 

2. 5j/-6 = 9j + 42 

3. -6£ + 11 = 2^ + 43 

4. x-'20 = 50-6x 

5 . -2^ + 10 = 4 

6. 10-3;tr = -20 

7. 6-4j=2 



8. 14 = 27 + 20 

9. -7# + 4=+8tf-41 

10. -2;r-7=-8;t--19 

11. + by - 3 - %y - 16 

12. 5 + 2^ = 

13. 10^ + 22 = 12 

14. = 4;r + 20 



Section 91. A short way to solve an equation : To 
TRANSPOSE terms from one side of the equation to the 
other. In solving the preceding equations you were often 



2oo Fundamentals of High School Mathematics 



forced to add or subtract something on each side of the 
equation. For example, see the illustrative example on 
page 199. This becomes Very laborious. The work may 
be reduced very much by using the following short method. 
Several examples will illustrate it clearly. 



The old way of thinking about 
the equation. 

1. x-7^5. 
Add 7 to each side, gives 

x- 7 + 7 = 5 + 7, 
or x = 12. 

2. 4x + 9=l. 
Subtract 9 from each side 
gives 

4* + 9- 9 = 1-9, 
or 4 x = - 8. 

3. _2x-12 = 5x-40. 1 
Add 12 to each side, gives 

_2x-12 + 12 = bx -40+12, 
or - 2 x = 5 x — 28, 
and then Subtract 5x 
from each side, giving 
-7x=-28. 



The short cut for doing it. 

1. x-7 = 5. 
Transpose the — 7, gives 

x = 5 + 7, 
or x = 12. 

2. 4x + 9 = l. 
Transpose the + 9, gives 

4x = 1-9, 
or 4x=— 8. 

3. -2x-12 = 5x-40. 
Transpose — 12, and + 5 x, 
gives 

- 2 x — 5 x = — 40 + 12. 
Collecting terms, 

_7x=-28. 



In the first example, we wanted to get rid of — 7 on 
the left side of the equation. By the short method, the 
— 7 is put on the other side as + 7. 

In the second example, we wanted to get rid of + 9 on 
the left side. The short method simply puts — 9 on the 
other side. 

In the third example, we wanted to get rid of — 12 on 
the left side. The short way of doing it is to put + 12 on 
the other side. We also wanted to get rid of 5 x on the 
right side ; the short way of doing it is to put — bx on the 
other side. 



The Further Use of the Simple Equation 201 



These examples show that any term in an equation may 
be transposed from one side of the equation to the other 
side, by changing its sign. Transposition is merely a 
device for shortening the labor required in adding or sub- 
tracting the same number on both sides of the equation. 

The student should solve the following example by the 
old method, to see which he likes the better. 



Solve : 


-3x-8 = 8x-30. 


Transposing, 


— Zx— 8x= — 30 + 8. 


Collecting, 


-11* =-22. 


Dividing by — 11, 


x = 2. 


Check : 


- 6 - 8 = 16 - 30. 




- 14 = - 14. 



EXERCISE 79 
PRACTICE IN SOLVING EQUATIONS BY TRANSPOSING TERMS 

Solve the first three equations by both methods : 

1. 7* + 34 = -4*-10 

2. 8^-11 = -2^+9 

3. 6r + 13 = -17 + ^ 

4. 12*+8 = -32+2* 

5. = 27 -G*-3 

6 . 4 J _9 7 = _2 J /-39 

7. 6^-11^ + 40 = 

8. 13/ - 27 = 20/ + 8 

9. 23* -82 = 30* + 2 
10. _12j/ = -18-6r 
u. 17 -3£ = - Sd-S 

12. 1-9/ = 9/ + 10 

13. = 2* -12 -14 



14. 


4^ = 16 + 6^: 


15. 


4*- 7 = 53 -6* 


16. 


1^4- 1— 1 x 
8 ■ T 6 2"* 


17. 


fj/ + 5 = 20 


18. 


H- 2 


19. 


***-' 


20. 


2*-8 4-* 
5 6 


21. 


2» + 3 , 3 

5 +n ~2 



22. 



*-17 = 2*-47 



23. = by - 18 + y 



202 Fundamentals of High School Mathematics 



II. HOW TO SOLVE EQUATIONS CONTAINING PARENTHESES 

Section 92. We saw in the last chapter that parentheses 
were used to indicate multiplication. Thus, to show that 
— 4 is to be multiplied by — 6, we use the parentheses, as 
follows: (—4) (—6). Multiplication is usually indicated 
in this way. Take this example to illustrate the way in 
which parentheses will be used in equations : 

Illustrative example. 

Solve the equation which states 
that the perimeter of the rectangle 
in Fig. 124 is 54 inches. 

Solution : 2 • x= length of the 
two altitudes. 

2(2 x - 3) — length of the two bases. 

Therefore, 2 x + 2(2 x - 3) = 54. 

Note the use of parentheses, i.e. to show that the expression 
2 x — 3 must be multiplied by 2. 

(1) Removing parentheses, gives 

2x + 4x-6 = 54. 

(2) Transposing, 2x + 4x = 54 + 6. 

(3) Collecting, 6 x — 60. 

(4) Whence x = 10, the altitude 
and 2x — 3 = 17, the base. 




Fig. 12-1 



EXERCISE 80 
PRACTICE IN SOLVING EQUATIONS WHICH CONTAIN PARENTHESES 

Solve and check each of the following equations : 

1. 2O + 10)= 42 3. 3(2 £-4) =18 

2. 50-2)= 15 4. 4.r + 5<> + 2) = 46 



The Further Use of the Simple Equation 203 

5. 2(^r-3)+3(^-2)=8 7. -3^ + 6(^-4)= 9 

6. 5£ + 2(4-£)= 32 a -7£ + 4(2£-3)=16 

Section 93. Note that in all the foregoing examples the 
number before the parenthesis has been positive. If nega- 
tive numbers occur, however, we proceed just the same, 

remembering how to multiply a negative number. 

Illustrative example. 

Solution of an equation involving REMOVAL OF PARENTHESES. 

9. %x-2(2x-7)= jc+8. 
The expression 2 x — 7 is to be multiplied by — 2. 

(1) Performing this multiplication, or removing parentheses, 
gives 

8x-4x + 14 = x + 8. (Why is it + 14 ?) 

(2) Transposing, gives 

Sx — 4x — x = 8 — 14. 

(3) Collecting, 3 x = - 6. 

(4) Dividing each side by 3 gives 

x=-2. 

(5) Check : Substituting — 2 for x throughout the equation 
gives 

_16-2(-4-7) = -2+8. 
- 16 + 8 + 14 =-2 + 8. 
6 = 6. 

EXERCISE 80 (continued) 

io. 5^-3(4- 2ff)= 2 £ + 42 

11. 6(^-3)-4 + (-r2)=4-^ 

12. 7(£-2)-2(3 + £)=0 

13. 4(2j/-5)+15 = 3(j/ + 10) 

14. 9j-3(2j-4)=6 



204 Fundamentals of High School Mathematics 



Section 94. A difficult form o( multiplication. A form 
of multiplication that gives pupils difficulty is the kind 
represented by — (4 — 5;r) in the equation : 
15. 5 -2(*-6)= -(4 -5x) 
When no multiplier appears immediately before the 
parentheses, the multiplier 1 is understood. Therefore in 
this case, it is just as though the right side of the equa- 
tion read 

-1(4-5*).. 

Illustrative example. 

The complete set of steps required to solve this equa- 
tion includes : 

(1) Removing parentheses gives 

5-2x + 12=-4 + 5x. (Why is it +5 x?) 

(2) Combining terms gives 

-2x + 17=-4 + 5x. 

(3) Transposing gives 

_2jc-5x=-4-17. 

(4) Collecting, 

-7x=- 21. 

(5) Dividing each side by — 7 gives 

x = Z. 

(6) Substituting 3 for x, throughout the original equation to 
check the result, gives 

5_2(3-6) = -(4-5-3). 
or, 5 -6 + 12 =-4 + 15. 

11 = 11. 

There are two important and difficult points in this last 
example. First, you should note that in the expression 
5 — 2{x — 6) the — 2 is not to be subtracted from the 5. 
The expression in parentheses must be multiplied by — 2. 
Second, if no multiplier is written before the parentheses, 
as in the expression — (4 — 5*), it is understood that the 
multiplier is 1. In this case it is — 1. If there had been 



The Further Use of the Simple Equation 205 



no sign before the parentheses, as (4 
would be understood to be 4- 1. 



5x), the multiplier 



EXERCISE 80 (continued) 

16. 7x-(x-±)=25 21. 2b-l(S-b)=b+8 

17 . _5j/-(2-j)=18 22. 1(2*+ 3)= -IT 

18. 6*-{*+7)=-2*+35 23. -1(6 -2*)= 43 

19. 5 - 2(* - 4)= 23 24. -(6-2*)= 24 

20. 7 -12(3-£)= 31 25. 16= (2* + 4) 

A SUMMARY OF THE STEPS REQUIRED TO SOLVE 
EQUATIONS WHICH CONTAIN PARENTHESES 

Section 95. Look back to the illustrative examples, 9 and 
15, and compare the steps in the solution that is worked out 
for each one with the steps in the solution in each of those 
which you have just worked. You will note that to solve 
such an equation the following steps are always included: 
I. Removing the parentheses (i.e. multiplying). 
II. Combining like terms on each side. 

III. Transposing to get all the knowns on one side and 
all the unknowns on the other. 

IV. Dividing each side by the coefficient of the unknown, 
to give the numerical value of the unknown. 

V. Substituting the obtained value in the original equa- 
tion to check the result. 



REVIEW EXERCISE 81 



+ *=18 



4. *-2Or-4)=0 



2. | /+ 4 J = 2 5. j/-(10-j/)-12 = 2 

3. 3(^-5)-*= -23 6. 2c = 4:c-16 

7 . -(^-6)-3^=2(^r-12) 



206 Fundamentals of High School Mathematics 

-2. 



10. 



b__Zb 

2 5 

3 

n 9 

2£ = 8 

5 3 



8. - — 



9. _=- 



11. -(20-3j/)=l 

12. .2j/-.5j = 120 



13. .2(6£-l)=2.6 

14. 2^-8^-20= -4^+42 

15. - 14 - 10j/ = + ±y - 84 
STANDARDIZED PRACTICE EXERCISE C (TIMED) 

Practice until you can reach the "standard," 12 exam- 
ples right in 5 minutes. 



1. 5-2^=13-4^- 

2 . 7£-4 + 6 = 3£-10 + 2 

3. 5^ + 3 = -2 

4. 4tf + 8 = 6tf-5 ..." 

5. 2j/-5 + 8j = 11 + 7j/+2 

6. 3 = 4,r + 9 . 

7. 10-j/=3j/ + 18 

8. 13-5^+7 = ^+3-2^ 

9. 5 £ - 7 = 8 

10. -20-36 = 7^ + 9 

11. 11^-7-4^ = 5^ + 8-19 

12. 9 = 4;r+7 

13. 8-4tf = 12-6^ 

14. 9 + ^-6x=5^r+3 +21 

15. 12^ + 3= -21 . 

16. 21-4a = 17+tf 

17. -2^-8 + 5^= + 2^r+7 + 5 ... 

18. 1 x = — 12 — x 



The Further Use of the Simple Equation 207 

SECOND TASK OF THE CHAPTER 

B. A FURTHER STUDY OF HOW TO SOLVE WORD 
PROBLEMS ALGEBRAICALLY 

Section 96. Review of important steps in translating 
word problems into algebraic statements. We have taken 
a great deal of time to learn how to solve any kind of 
simple equation because we need to be able to use equa- 
tions skillfully in solving actual problems later. The 
problems as a rule will not be stated for us, in algebraic or 
equational form, all ready for solution. They will be 
stated merely in words. First, then, we shall always have 
to translate the word problem into an equation. After this 
first step the work is the mere solution of the equation. 

Our second principal task in this chapter, therefore, 
is to become skillful in translating word problems into 
algebraic form. We learned in our work with Chapter II 
the important steps in translating word problems. Since 
we are to learn in the next few lessons how to translate a 
great many different kinds of word statements, let us re- 
view these steps here : 

First step : See clearly which things in the prob- 
lem are known and which are un- 
known. 
Second step : Represent one of the unknowns, most 
conveniently the smallest one, by some 
letter. 
Third step : Represent all of the others by using 

the same letter. 
Fourth step : By careful study of the relations 
between the parts of the problem, ex- 
press the word statement in algebraic 
form. 



208 Fundamentals of High School Mathematics 

Sometimes this will mean an equation and sometimes not. 

For the next few lessons, therefore, you will work many- 
word problems. The exercises are included to give prac- 
tice in translating many different kinds, so that you will be 
able to use the method in solving any kind that you may 
happen to meet later. For convenience they will be 
arranged by types, examples of the same type being 
studied together. 

Section 97. Need for tabulating the data of word prob- 
lems. Many problems involve so many different state- 
ments that it is practically necessary to arrange the steps in 
the translation in very systematic tabular form. Take an 
example like this : 

John's age exceeds James's by 20 years. In 15 years he will be 
twice as old as James. Find the age of each now. 

Before we can write this statement in the form of an equa- 
tion we must express in algebraic form four different things : 
(1) John's age now ; (2) James's age now ; (3) John's age 
in 15 years ; and (4) James's age in 15 years. These 
four facts can best be stated in a table like this : 

(First step) Let n represent James's age now. 

(Second step) Tabulate the data : 



Table 15 





Age now 


Age in 15 years 


John's age 


n + 20 


T1 + 20+15 


James's age 


n 


n +15 



With all the facts expressed in letters we can now state 
the equation which tells the same thing as the original word 
statement ; namely : 



The Further Use of the Simple Equation 209 

(Third step) n + 20 + 15 = %n + 15). 

We are now ready for the 
(Fourth step) the solution of the equation ; the steps are 

as follows : 

(1) n + 35 = 2 n + 30. 

(2) . - n = - 5. 

(3) n = 5. 

Therefore James's age now is 5, and John's age now 
is n + 20, or 25. 

(4) Check the accuracy of this result thus : 

In 15 years John will be 40 and James will be 
20 ; or John will be twice as old as James, as 
the problem states. 
To be proficient in solving such problems, therefore, we 
first need practice in tabulating such facts as " age now," 
" age some other time," as in this example. Other types 
which involve the same need for tabulation will be taken 
up later. 

I. PROBLEMS RELATING TO AGE 

EXERCISE 82 

PRACTICE IN REPRESENTING RELATIONS BETWEEN NUMBERS 

1. A man is now 25 years of age. What expression 
will represent his age : 

(a) 10 years ago ? (c) x years ago ? 

(b) 8 years from now ? (d) m years from now? 

2. C is now n years of age. What expression will 
represent his age: 

(a) 12 years from now ? (c) y years ago? 

{b) 7 years ago? (d) m years from now? 



2io Fundamentals of High School Mathematics 



3. A is now x years old. B's present age exceeds 
A's age by 8 years. What expression will repre- 
sent : 

id) B's present age ? 

(V) the sum of their ages ? 

(c) the age of each 10 years ago? 

(d) the age of each 5 years from now ? 

(e) the sum of their ages in 5 years ? 

4. A is now n years of age ; B is three times as old. 
Express algebraically : 

(a) B's present age ; 

(b) the age of each 4 years ago ; 

(c) the age of each 9 years from now. 

(d) State algebraically that B's age 4 years ago 
was 5 times A's age then. 

5. A's present age exceeds B's present age by 25 
years. In 15 years he will be twice as old as B. 
Find their present ages. 

6. C is six times as old as D. In 20 years C's age 
will be only twice D's age 20 years from now. 
What are their present ages ? 

7. A man is now 45 years old and his son is 15. In 
how many years will he be twice as old as his 
son ? 

8. A father is 9 times as old as his son. In 9 years 
he will be only 3 times as old. What is the age 
of each now ? 

9. A's present age is twice B's present age ; 10 
years ago A's age was three times B's age then. 
Find the age of each now. 



The Further Use of the Simple Equation 211 

II. PROBLEMS IN WHICH A NUMBER IS DIVIDED INTO 
TWO OR MORE PARTS 

Section 98. The solution of a great many problems 
depends upon our being able to separate a number into 
two or more parts. For example, if a man has a certain 
sum of money to invest, he may invest part of it in one 
thing, and part in another. The solution of such an ex- 
ample requires that we be able to divide a number into 
two or more parts algebraically. 

EXERCISE 83 
PRACTICE IN DIVIDING A NUMBER INTO TWO OR MORE PARTS 

1. The sum of two numbers is 20. 

(a) Express in algebraic form the second one if 
the first one is 12. 

(b) Express in algebraic form the second one if 
the first one is n. 

(c) Express in algebraic form the fact that the 
second one exceeds the first one by 4. 

2. There are 36 pupils in a mathematics class. 

(a) Express algebraically the number of boys if 
there are 19 girls. 

(b) Express algebraically the number of girls if 
there are n boys. 

(c) State algebraically that there were 6 more 
girls than boys. 

3. A farmer has two kinds of seed, clover seed and 
blue grass seed. If he has 100 lb. of both, 

express : 



212 Fundamentals of High School Mathematics 

{a) the number of pounds of clover seed if there 

were 24 lb. of blue grass seed ; 
(b) the number of pounds of clover seed if there 

were n lb. of blue grass seed ; 
(V) the value of the clover seed (n lb.) at 20 cents 

per pound ; and the value of the blue grass 

seed at 15 cents per pound. 
(d) State by an equation that the value of both 

kinds together was $19. 

4. Divide 20 into two parts such that the larger 
part exceeds the smaller part by 4. 

5. A boy paid 48 cents for 20 stamps ; some cost 
two cents each and the remainder cost three 
cents. How many of each kind did he buy ? 

6. During one afternoon a clerk at a soda fountain 
sold 200 drinks, for which he received $16. 
Some were 5 cents each ; the others were 10 
cents each. Find the number of each kind. 

7. A grocer has two kinds of coffee, some selling 
at 30 cents per pound and some selling at 
50 cents per pound. How many pounds of 
each kind must he use in a mixture of 100 
pounds which he can sell for 34 cents per 
pound ? 

III. PROBLEMS BASED ON COINS 

Section 99. Another illustrative type of word problem 
which gives practice in tabulating data and thus in solving 
difficult word problems is the " coin problem." Take this 
example : 



The Further Use of the Simple Equation 213 



Illustrative example. A man has 3 times as many 

dimes as quarters. 

How many of each has he if the value of both together is 

$11? 

Here there are four distinct numbers to be expressed, as in 

the case of the age problem: (1) the number of quarters; 

(2) the number of dimes ; (3) the value of the quarters in 

terms of a common base (for example, cents) ; (4) the value 

of the dimes in the same base (cents). The steps in the 

solution are clear, therefore, from the following illustrative 

solution : 

Table 16 





Number 


Value (cents) 


quarters 


n 


25 n 


dimes 


3n 


30 71 



(1) Let 

(2) Then 
(3) 
(4) 
(5) 

(«) 
(&) 



n = the number of quarters. 
3 n = the number of dimes. 
25 n+ 30n= 1100 cents. 

.-. n = 20, number of quarters. 
3 n = 60, number of dimes. 
Value of the quarters = 95. 
Value of the dimes = $ 6. 
Total value = 9 11, as stated in the example. 



EXERCISE 84 
PRACTICE IN EXPRESSING THE VALUE OF VARIOUS NUMBERS OF COIHS 

1. Express the value in cents of : 

(a) d dimes; (d) 4 d half dollars ; 

{b) 3 d quarters ; (e) d dollars ; 

(c) 2 d nickels ; (/) of all the coins. 

2. Express the value in cents of: 

(a) n nickels ; (c) (n + 5) quarters; 

(b) (3 - n) dimes ; \d) (12 - n) half dollars; 

0) (30 - n) nickels. 



214 Fundamentals of High School Mathematics 

3. A purse was found which contained nickels and 
dimes, 20 in all. Find the number of each if the 
value of both was $1.60. 

4. I received at a candy counter twice as many dimes 
as quarters, and 6 more nickels than dimes and 
quarters together. How many of each coin did 
I receive if the value of all was $7.50 ? 

5. A debt of $ 72 was paid with 5-dollar bills and 
2-dollar bills, there being twice as many of the 
latter as of the former. Find the number of 
each kind of bill. 

6. 18 coins, dimes and quarters, amount to $2.25. 
Find the number of each kind of coin. 

7. A cab driver received twice as many quarters as 
half dollars, and three times as many dimes as 
half dollars; in all he had $13. How many of 
each coin did he receive? 

IV. PROBLEMS BASED ON TIME, RATE, AND DISTANCE 

Section 100. In Chapter VIII we saw that the motion of 
a train could be represented graphically. Now we shall 
learn how to solve this kind of problem by means of the 
equation. 

EXERCISE 85 

PRACTICE IN SOLVING PROBLEMS BASED ON RELATIONS BETWEEN TIME, 
RATE, AND DISTANCE 

1. Express the distance covered by an automobile 
in 10 hours if its rate is : 

(a) 18 miles per hour ; 

(b) 5 miles per hour ; 



The Further Use of the Simple Equation 215 

(c) (r-h 3) miles per hour ; 

(d) ("2x— 5) miles per hour. 

x. A train runs for / hours. Express the distance 
it will cover at the rate of : 

(a) 35 miles per hour ; 

(b) m miles per hour ; 

(c) (r+ 6) miles per hour ; 

(d) t miles per hour. 

3. An automobile tourist sets out on a 400-mile 
trip. Express the time required if he goes at the 
rate of : 

{a) 40 miles per hour ; 
{U) 5 miles per hour ; 

(c) (r + 10) miles per day ; 

(d) (2 r — 3) miles per day. 

4. How long will it require to make a trip of D 
miles at the rate of 15 miles per hour ? 5 miles 
per hour ? 

5. At what rate must one travel to go D miles in 
10 hours ? In t hours ? In t + 3 hours ? 

6. A slow train travels at the rate of 5 miles per 
hour ; a fast train travels 15 miles more per 
hour. Express : 

(a) the rate of the fast train ; 

(b) the distance passed over by each in 5 hours. 

(c) State algebraically that the two trains to- 
gether traveled 100 miles in 5 hours. 

7. Two trains leave Chicago at the same time, one 
eastbound, the other westbound. The east- 
bound train travels 10 miles less per hour than 
the westbound train. Express : 



216 Fundamentals of High School Mathematics 

(a) the rate of each ; 

(b) the distance traveled by each in 4 hours, 
(r) Form an equation stating that they were 

440 miles apart at the end of 4 hours. 

8. Two trains, 350 miles apart, travel toward each 
other at the rate of 40 and 35 miles per hour, 
respectively. 

(a) Express the distance traveled by each in / 
hours. 

(b) Form an equation stating the fact that the 
trains met in t hours. 

9. Make formulas for d> for /, and for r, that can 
be used in any problem based upon uniform 
motion. 

10. Illustrative example. Two bicyclists, 200 miles apart, 
travel toward each other at rates of 12 and 8 miles per hour 
respectively. In how many hours will they meet ? 
(1) Let f represent the number of hours until they meet. 



(2) 


Table 17 






Time in 
hours 


Rate per hr. 
in miles 


Distance in 
miles 


For slow one 
Rr fast one 


t 
t 


8 
12 


8t 

12 1 



(3) Then 8 1 + 12 1 = 200. 

(4) .-. * = 10. 

11. Two men start from the same place, one going 
south and the other going north. One goes 
twice as fast as the other. In 5 hours they are 
120 miles apart. Find the rate of each. 



The Further Use of the Simple Equation 217 

12. An eastbound train going 30 miles per hour left 
Chicago 3 hours before a westbound train going 
36 miles per hour. In how many hours, after 
the westbound train left, will they be 519 miles 
apart ? 

13. A bicyclist traveling 15 miles per hour was over- 
taken 8 hours after he started by an automobile 
which left the same starting point 4J hours later. 
Find the rate of the automobile. 

14. A starts from a certain place, traveling at the 
rate of 4 miles per hour. Five hours later B 
starts from the same place and travels in the 
same direction at the rate of 6 miles per hour. 
In how many hours will B overtake A ? 

REVIEW EXERCISE 86 

1. The sum of two numbers is 55; twice the 
greater equals three times the .smaller, plus 15. 
Find each number. 

2. A is 10 years older than C, and B is 6 years 
younger than C. The sum of their ages six 
years ago was 40 years. How old is each now ? 

3. A is now 17 years old and B is 50. In how 
many years will A be exactly one half as old as 
B? 

4. The value of 36 coins, dimes and quarters, is 
$ 6.60. Find the number of each kind of coin. 

5. A collection of nickels, dimes, and quarters 
amounts to $4. There are 10 more nickels 
than dimes, and 2 less quarters than dimes. 
Find the number of each. 



2i8 Fundamentals of High School Mathematics 

6. One car, running 20 miles per hour, left a cer- 
tain point 4 hours after another car which was 
running at the rate of 15 miles per hour. In 
how many hours will it overtake the other one ? 

7. A and B start from the same place, and travel 
in opposite directions. A's rate is twice B's rate. 
In 4 hours they are 120 miles apart. Find the 
rate of each. Make a drawing to illustrate. 

8. A leaves a certain place at 10 o'clock, traveling 
due north. B starts from the same place at 
12 o'clock, and travels due south. At 4 o'clock 
they are 210 miles apart. Determine how fast 
each traveled, if A's rate is one half of B's rate* 
Make a drawing. 

9. A left a town 4 hours after B left. They trav- 
eled in opposite directions, A at the rate of 
12 miles per hour, and B at the rate of 20 miles 
per hour. In how many hours will they be 
272 miles apart ? 

10. A freight train left Kansas City for St. Louis 
at the rate of 12 miles per hour at the same 
time that a passenger train running 45 miles 
per hour left St. Louis for Kansas City. How 
long will it be before they meet, if the distance 
between the two cities is 285 miles ? 

11. Two hours after a messenger, who was traveling 
at the rate of 10 miles per hour, left a camp, it 
was decided to cancel his message. How fast 
must a second messenger travel to overtake him 
in 8 hours ? 



The Further Use of the Simple Equation 219 

12. Two autoists 200 miles apart start toward each 
other. The faster one travels 20 miles per hour 
and the slower one 15 miles per hour. In how 
long will they meet, if the faster one is delayed 
2 hours on the trip ? 

V. PROBLEMS INVOLVING PER CENTS 

Section 101. Many problems involving per cents may be 
solved by algebraic methods. 

EXERCISE 87 
PRACTICE IN SOLVING PERCENTAGE PROBLEMS 

1. What does 10 % mean ? 5 % ? r% ? 

2. Indicate 4 % of 8600 ; 5 % of $275. 

3. Express decimally 5 % of / ; 8 % of c • 6J% of b. 

4. A man paid c dollars for an article. He sold it 
at a gain of 25 %. Express : 

(a) the gain in dollars ; 

(b) the selling price. 

(c) State algebraically that he sold the article 
for 82.50. 

5. A merchant sold a suit for $25, thereby gaining 
25%. If the cost is represented by c dollars, 
what will represent : 

{a) the gain in dollars ? 

(b) the selling price in terms of c? 

(c) State algebraically that the selling price was 
825. 

6. Solve each of the following equations: 

(a) .20^=180 

\b) x + .06*= 3.18 



22o Fundamentals of High School Mathematics 



7. 



8. 



(c) c + . 10 c = 495 

(d) m - .15 m = 21.25 

(e) p -f- .04/ = 520 
if) x- .50 x = 18.75 
(/) 2 - %x - .5x= 7 
(k) 1.75 x- J* =1000 

Find the cost of an article sold for $ 156 if the 
gain was 10 %. (Use c for the cost.) 
What number increased by 66| % of itself 
equals 150 ? 

After deducting 15 % from the marked price of 
a table, a dealer sold it for % 21.25. What was 
the marked price ? 

A dealer made a profit of $3690 this year. 
This is 18 % less than his profit last year. Find 
his profit last year. 

A number increased by 12.5 % of itself equals 
243. What is the number ? 
A shoe dealer wishes to make 25 % on shoes. 
At what price must he buy them in order to 
sell them at $4.50 per pair ? 
A furniture dealer was forced to sell some dam- 
aged goods at 14 % less than cost, and sold 
them for $ 129. How much did they cost ? 
A man sold a suit of clothes for $30.25. What 
per cent did he gain if the clothes cost him 
$25? 

VI. INTEREST PROBLEMS 

Section 102. Many interest problems can be more easily 
solved by algebraic equations than by the methods of 
arithmetic. 



10. 



ii. 



12. 



13. 



14. 



The Further Use of the Simple Equation 221 

EXERCISE 88 

1. Express the interest on 1 150 at 5 % for 1 year; 
for 3 years ; for / years. 

2. Express the interest on P dollars at 6 % for 1 
year ; for 3 years ; for t years. 

3. Express the simple interest on $ 500 for 1 year 
at r °lo \ for 4 years. 

4. A man borrowed a certain sum of money at 6 %. 
Express : 

(a) the interest for 2 years. 

(b) State algebraically that the interest for three 
years was $48. 

5. What principal must be invested at 6 % to yield 
an annual income of $ 57 ? 

6. For how many years must $ 2800 be invested 
at 7 % simple interest to yield $ 833 interest ? 

7. What is the interest on P dollars at r°]o for t 
years ? 

8. A man invests part of % 1000 at 4 %, and the 
remainder at 6 %. If x represents the number 
of dollars invested at 4 %, express : 

(a) the annual income on the 4 % investment; 

(b) the amount of the 6 % investment ; 

(c) the annual income on the 6 % investment. 

(d) State algebraically that the annual income 
on the 4 % investment exceeds the annual 
income on the 6 % investment by $ 20. 

9. Part of $ 1200 is invested at 5 % and the re- 
mainder at 7 %. The total annual income from 
the two investments is 167. What was the 
amount of each investment ? 



222 Fundamentals of High School Mathematics 

10. Ten thousand dollars' worth of Liberty Bonds 
yield an annual interest of $ 370. Some pay 
3^ %, and the remainder pay 4 %. Find the 
amount of each kind of bond. 

11. A 5 % investment yields annually $ 5 less than 
a 4 % investment. Find the amount of each 
investment if the sum of both is $ 800. 

VII. PROBLEMS CONCERNING PERIMETERS AND AREAS 

Section 103. The following examples are based on 
squares and rectangles : 

EXERCISE 89 

1. The length of a rectangle exceeds twice its width 
by 12 in. Represent its width by w. 

(a) Make a drawing to represent it. 
. (#) Express its length. 

(c) Express its area. 

(d) Express its perimeter. 

(e) State that its perimeter is 84 in. Solve the re- 
sulting equation. 

2. The length of a rectangle is 9 in. more, and the 
width is 6 in. less, than the side of a square. 

(a) Make a drawing for each. 

(b) Express the dimensions of the square. 

(c) Express the dimensions of the rectangle. 

(d) Express the perimeter of the rectangle. 

(e) Express the area of the rectangle. 

(/) State algebraically that the sum of the perim- 
eters is 168 in. Solve the resulting equation. 

3. The base of a triangle exceeds its height by 10 
inches. 



The Further Use of the Simple Equation 223 

{a) Make a drawing for the figure. 

(b) Express its base and height. 

(c) Express its area. 

(d) State that its area is equal to the area of a 
rectangle whose dimensions are 8 in. and 
5 in. (Note that you have not yet learned 
how to solve equations like this. We will 
take them up in Chapter XVIII.) 

The length of a rectangle is 4 feet more, and 

its width is 2 feet less, than the side of a square 

whose perimeter is P inches. Express : 

(a) the side of the square ; 

(3) the dimensions of the rectangle; 

(c) the perimeter of the rectangle. 

(d) Find the value of P if the perimeter of the 
rectangle is 44 inches. 

A tennis court for singles is 3 feet shorter than 
3 times its width. Find the length and width of 
the court, if its perimeter is 210 ft. 

The width of a basket-ball court is 20 ft. less 
than its length. The perimeter is 240 ft. Find 
its dimensions. 

A picture, twice as long as it is wide, is inclosed 
by a frame an inch wide. The perimeter of the 
outer edge of the frame is 44 inches. What is 
the size of the picture ? 

The length of a rectangle exceeds its width by 
10 inches. If each dimension is increased by 
5 inches, the resulting perimeter will be 128 
inches. Find the area of the original rectangle. 



224 Fundamentals of High School Mathematics 

9. If P equals the perimeter of a square, each of 
whose sides is s, what will be the perimeter of a 
square each of whose sides is 3 s ? 

VIII. PROBLEMS BASED ON LEVERS 

Section 104. A teeter board is one form of lever. The 
point on which the board rests or turns is the fulcrum ; 
the parts of the board to the right of and to the left of the 
fulcrum are the lever arms. 




Fig. 125 

If a boy at A, who just balances a boy at B, moves to 
the left while B remains stationary, it is clear that the left 
side goes down. But if the boy at B moves closer to the 
fulcrum while A remains stationary, then A goes down. 
It is also clear that boys of unequal weight cannot teeter 
unless the heavier boy sits closer to the fulcrum. There 
is a mathematical relation between the weight on the lever 
arm and its distance from the fulcrum. Two boys will 
balance each other when the weight of one times his dis- 
tance from the fulcrum is equal to the weight of the other 
times his distance, or in general, when 

weight times distance on one side equals weight times 
distance on the other side. 

This law or relation may be tested by placing equal coins 
at different positions on a stiff ruler balanced on the edge 
of a desk. Try this experiment. See whether 2 pennies 



The Further Use of the Simple Equation 225 

placed 4 inches from the fulcrum (at the center of the lever) 
will balance 1 penny placed 8 inches from the fulcrum. 
See whether 6 pennies placed 2 inches from the fulcrum 
will balance 3 pennies placed 4 inches from the fulcrum. 

Thus, to make a lever balance, the product of weight 
and distance from the fulcrum on one side must equal the 
product of weight and distance from the fulcrum on the 
other side. 

EXERCISE 90 
PROBLEMS BASED ON LEVERS. MAKE A DRAWING FOR EACH: 

1. John weighs 80 lb. and sits 4 ft. from the ful- 
crum. Where must Robert sit if he weighs 90 lb. ? 
\ 2. A, weighing 120 lb., sits 4^ ft. from the fulcrum, 
and balances B, who sits 5 ft. from the fulcrum. 
What is B's weight ? 

3. A hunter wishes to carry home two pieces of 
meat, one weighing 40 lb. and the other 60 lb. 
He puts them on the ends of a stick 4 ft. long 
and places the stick across his shoulder. Where 
must the fulcrum (his shoulder) be placed to 
make the weights balance ? 

4. Two children play teeter, one on each end of a 
board 9 ft. long. Where must the fulcrum be if 
the children weigh 60 and 80 lb. respectively ? 

5. Could three children teeter on the same board ? 
How ? 

6. A and B sit on the side of the fulcrum. A 
weighs 100 lb. and sits 5 ft. from the fulcrum ; B 
weighs 80 lb. and sits 3 ft. from the fulcrum. 
Where must C sit to balance the other two, if he 
weighs 150 lb. ? 



226 Fundamentals of High School Mathematics 



SUMMARY OF CHAPTER X 

This chapter has taught all the steps involved in solving 
a simple equation : 

1. Removal of parentheses. 

2. Getting rid of fractions. 

3. Transposing to get all the knowns on one side 
and all the unknowns on the other. 

4. Dividing each side by the coefficient of the un- 
known. 

5. Checking by substituting the obtained value of 
the unknown in the original equation. 

Many kinds of word problems have been solved. Tabu- 
lating the information or data of such problems is a great 
help in solving them. A systematic method always pays big 
dividends in any kind of work. 

EXERCISE 91 
MISCELLANEOUS PROBLEMS 

1. A grocer has two kinds of tea, — some worth 
60 ^ per pound and some worth 75 <f, per pound. 
He has 20 lb. more of the former than of the 
latter kind. How many pounds of each kind 
has he, if the value of both kinds is $45.75? 

2. I bought 45 stamps for $1.05. If part of them 
were 2-cent stamps and part 3-cent stamps, 
how many of each did I buy ? 

3. The sum of the third, the fourth, and the eighth 
parts of a number is 17. What is the number ? 



The Further Use of the Simple Equation 227 

4. John has | as many marbles as Harry. If John 
buys 120 and Harry loses 23, John will then 
have 7 more than Harry. How many has each 
boy ? 

5. A clerk spends \ of his yearly salary for board 
and room, \ for clothes, ^ for other expenses, 
and saves $ 880. What are his annual expenses ? 

6. A father left one third of his property to his 
wife, one fifth to each of his three children, and 
the remainder, which was $1200, to other rela- 
tives. Find the value of his estate. 

7. Ten years ago A was one third as old as he is 
at present. Find his age now. 

8. Find C in the formula C= ^ F ~ 82 ^ if F= 20. 

y 

9. A merchant bought goods for $500, and sold 
them at a gain of 5%. What was the selling 
price ? 

10. If in problem 9 the merchant had sold the 
goods at a gain of x per cent, what would have 
been the selling price ? 

11. A 6-foot pole casts a shadow 4|- ft. in length. 
At the same time how long is the shadow of an 
8-foot pole ? 

12. The ratio of two numbers is |. Find each 
number if their sum is 56. 

13. Two numbers differ by 70 ; the ratio of the 
larger to the smaller is \. Find each number. 



228 Fundamentals of High School Mathematics 



14. In Fig. 126, ZC= 

90°, A A = 37°, and 
AC =24. Find AB, 
BC, and Z.B. jg 




Fig. 126 



15. In general, which is the larger, the cosine or the 
tangent of an angle ? Show by a drawing. 

16. The highest office building in the world (the 
Woolworth Building, New York City) casts a 
shadow 1240 ft. long at the same time that a 
boy 5 ft. tall casts a shadow 8 ft. long. What 
is the height of the building ? 

17. The table below gives the annual cost of pre- 
mium per $ 1000 life insurance, at various ages. 



Age 


21 


25 


30 


35 


40 


45 


50 


Premium 


18,25 


20.04 


22.60 


26.40 


30.50 


36.10 


45.20 



Show this graphically. Measure age along the 
horizontal axis. What would probably be the 
cost at age 28 ? at 42 ? 

18. In how many years will $450 earn $67.50 in- 
terest at 6 % ? 

19. If you know an acute angle, and one side, of a 
right triangle, can you find either of the other 
sides ? Illustrate. 

20. Which is the greater, 5x 2 or (5x) 2 ? 



CHAPTER XI 

HOW TO SOLVE GRAPHICALLY EQUATIONS WHICH 
CONTAIN TWO UNKNOWNS 

Section 105. Importance of skill in drawing the picture 
or graph of an equation. In Chapter VIII we learned how 
to represent and to determine the relationship between 
quantities that change together. Three methods of doing 
this were studied : (1) the tabular method ; (2) the graphic 
method ; (3) the equational or formula method. One of 
the most important facts for us to recall is that the graph 
and the equation tell exactly the same thing. For exam- 
ple, on page 151, the line BC and the equation C ' = .12 n 
tell exactly the same thing. Any information that you 
get from the equation you can also get from the graph. 
Furthermore, relationsliips can be seen more easily from 
graphs than from tables or equations. For these reasons, 
and since much of our later work in mathematics makes 
use of graphic methods, we need to be skillful in drawing 
the line which stands for an equation. 

Section 106. We need to know how to locate or to " plot " 
points. But a line may be regarded as a series of points. 
Thus, to represent or locate a line we have to locate a 
series of its points. It happens that much of our elemen- 
tary work, furthermore, deals with straight lines. This 
kind of line, clearly, can be fully determined by locating 
any two of its points. 

HOW TO LOCATE OR PLOT POINTS 

Thus, we see that the important thing in "graphing" 
is how to locate, or to represent, points. In every graph 
that you have already constructed you have had to locate 
points through which to draw the line. For example, in con- 
structing a cost graph it is necessary to locate several points 

229 



230 Fundamentals of High School Mathematics 

representing the cost of different numbers of units of the 
article. Let us study more carefully how points are located. 

Section 107. How points are located on maps. (1) Points 
are located on maps by means of latitude and longitude. 
Any point on the earth's surface is definitely located by 
stating its distance east or west of the'prime meridian, and 
its distance north or south of the equator. 

Thus, to the nearest degree, the location of New York is 
74° W. and 41° N. because it is 74° west of the prime meridian 
and 41° north of the equator. Similarly, the position of 
Chicago is 88° W. and 42° N. ; that of Paris, 2° E. and 49° N. 

(2) This same method is used by many cities in number- 
ing their houses. Two streets, which make right angles 
with each other, are selected as reference streets. Any 
house or building is completely located, then, by stating 
the number of blocks it is east or west, and north or south, 
of these reference streets. 

Section 108. How points are located on drawings. By a 
method similar to that above, we locate points on paper. 
Instead of using the equator and the prime meridian as 
our reference lines, we take two lines, — for convenience, 
one horizontal and the other vertical, — which make a right 
angle with each other. Any point may be located, then, 
by stating its distance to the right of, or to the left of, the 
vertical reference line ; and its distance above or below the 
horizontal reference line. 

Thus, in Fig. 127, point A is 2 units to the right of, 
and 1 unit above, the reference lines ; point B is 2 units 
to the left of, and 8 units below, the reference lines ; point 
C is 3 units to the left of, and 2 units below, the reference 
lines ; and point D is units to the right or left of, and 
3 units above, the reference lines. 



Solving Equations with Two Unknowns 231 
+Y" 

































P 












































A 

















































c 






















B 

























































X 



-Y 

Fig. 127 

Section 109. The point from which distances are meas- 
ured : The origin. The point in which the two axes meet, 
or their intersection point, is called the origin. It is the 
point from which we measure distances, either way. The 
origin is usually lettered with a capital O, as in Fig. 127. 

Section 110. How distances are distinguished from each 
other. It would be laborious to state that a particular 
point is " to the right of " or " to the left of " some refer- 
ence line, each time we refer to it. To avoid this, it has 
been agreed to call distances to the right of the Y-axis 
positive, and distances to the left of the Y-axis negative. 
Similarly, distances above the X-axis are positive, and dis- 
tances below it are negative. It is very important to remem- 
ber these facts, as we use them so often in graphic work. 

Thus, in Fig. 127, the position of point A is described by 



232 Fundamentals of High School Mathematics 



the numbers + 2 and + 1, or by (2, 1). This means that 
point A is 2 units to the right of the F-axis, and 1 unit 
above the X-axis. Similarly, the position or location of 
point B is described by the numbers — 2 and — 3, or by 
(—2, — 3); this means that point B is 2 units to the left of 
the F-axis and 2 units above the X-axis. In the same 
way, the position of point C is described by the numbers 
— 3 and — 2, or (— 3, — 2); this means that point C is 3 
units to the left of the F-axis and 2 units below the X-axis. 

At this time the student should note that in stating the 
location of a point, its distance to the right of, or to the left 
of, the F-axis is always given before its distance above or 
below the X-axis. This is done to avoid confusion. That 
is, the x-distance is always first, the j^-distance second. 
Remember that the "^-distance" means the distance to 
the right or the left of the vertical axis. 

Section 111. Plotting a point. By " plotting a point " we 
mean the locating, on cross-section paper, of a point whose 
;r-distance and ^-distance are known. 



Thus, to plot A, 
whose ^--distance is 
-f- 3 and whose ^-dis- 
tance is -+- 4, usually 
written (3, 4), means 
to locate on the cross- 
section paper a point v 
3 units to the right 
of, and 4 units above, 
the origin, as in Fig. 
128. In the same 
way, the point ( — 2, 1) 
is point B on the 
graph. 



+Y 



]E:E:E::: 



-Y 

Fig. 128 



Solving Equations with Two Unknowns 233 

EXERCISE 92 
PRACTICE m PLOTTING POINTS 

1. The ^--distance of a point is + 3, i.e. it is 3 units 
to the right of the vertical axis. Is it definitely 
located ? Why ? 

2. The ^-distance of a point is — 4, i.e. it is 4 units 
below the horizontal axis. Is it definitely located ? 
Why ? 

3. A certain point is on both axes. What are its x- 
andj-distances? 

4. Plot the points whose position is determined by 
the following : (4, 2), (5, 6), (- 3, 2), (- 4, - 1), 
and (-6, 2). 

5. Plot the following : (2, 8), (3, 7), (4, 6), (5, 5), 
(6, 4), (8, 2), (10, 0). 

6. Plot the following : (12, - 2), (15, - 5), (18, - 8), 
(10, 0), (5, 5). 

7. Plot the following : (21 3), (If, 5), (-21 3). 

HOW TO DRAW THE GRAPH OF AN EQUATION WHICH 
CONTAINS TWO UNKNOWNS 

Section 112. The picture of an equation. Now that we 
have learned how to locate, or plot, points, we come to the 
main purpose of the chapter : to show how equations can 
be solved graphically. 

First illustrative example. Let us take an equation 
which contains two unknowns, such as, 

y=2x + Z. 
In this equation the value of y changes as the value of x 
changes. Clearly, the value of y depends upon the value of %. 
For example, if x — T, then y — 5 ; if x = 2, then y = 7, etc. 
A table will help to show this relation between the unknowns, 
x and y. 



234 Fundamentals of High School Mathematics 
The equation isj/ = 2x + 3. 

Table 18 



If x equals 


1 


2 


3 


4 


5 


O 


-3 


-2 


-3 


~4 


"5 


then y equals 


5 


7 


9 


11 


13 


3 


1 


-1 


-3 


-5 


-7 



If we select any particular value of x, and the corresponding 
value of y which accompanies it, such as 1 and 5, or 2 and 7, 
we may think of them as completely describing the position 
of points on a graph. Thus, (1, 5), (2, 7), (3, 9), etc., 
definitely locate the position of the points. Plotting these points 
with respect to an X- and F-axis, we get a series of points, 
such as Fig. 129. By joining these points we obtain a straight 
line, which is the picture or the graphical representation of the 
equation y = 2x -f 3. 



+T 













22 




rt 




-ft7 




Zu 




-?/ 




A v 




























- - --4-TC 


tci 


— rA 







































-Y 

Fig. 129 



Solving Equations with Two Unknowns 235 

Second illustrative example. For a second example, 
let us consider an equation in two unknowns which shows 
that the sum of two numbers is always 10, such as x + y = 10. 
It is clear that one number, x, might be 2, and if so, that y 
must be 8 ; or that x might be 4, and if so, the other number, y, 
must be 6. Thus the two unknowns, x and y, may have many 
different values. A table helps to show this. 

The equation is x + y = 10. 

Table 19 



If x equals 


l 


2 


3 


4 


O 


-1 


-2 


-3 


then y equals 


9 


8 


7 


6 


10 


11 


12 


13 



Now we may think of any pair of related numbers, such as 1 
and 9, or 2 and 8, as describing the position of points on this 
line. Thus, (1, 9), (2, 8), (3, 7), (4, 6), etc., show the location 
of points on the graph. Plotting these points, we have 
Fig. 130. By joining these points we obtain in this illustra- 
tive example the "graph" or picture of the equation 
x + y = 10. Expressed in another way, we have represented 
graphically the relation between two numbers whose sum is 
always 10. 





1 


\ 










-S >F 




- s^* 




- ^% 




- s^ 




_N2I 








__ _ S, _-V 


r "^~ ~~ i" - 


S,- K 


_u j£ i 


s^ 




A 




























J_ + + 



-Y 



Fig. 130 



236 Fundamentals of High School Mathematics 



EXERCISE 93 

PRACTICE IN REPRESENTING GRAPHICALLY EQUATIONS WHICH CONTAIN 
TWO UNKNOWNS 

1. In the equation y = 2x + 5, find the value of y 
when x — 1 ; when x = 2 ; when x = 4 ; when 
x = ; when x = — 3. Make a table similar to 
the one above, showing these related pairs of 
numbers. 

2. In the equation x = y — 4, find the value of x 
when j/ = ; when j = 1 ; when y = 4 ; when 
jz = 6 ; when j/ = — 2 ; when jj/ = — 6. Tabu- 
late. 

3. Plot the equation given in Example 2. 



4. 



Plot the equation x + y = 6. Hint : First tabu- 
late related values of x and y. Use only four 
points. 

Plot, or graph, y = 5 + x. 
Plot 2x+y = 6, orj = 6 - 2;tr. 



7. Graph x — 2 j = 5, or x = 2 y -f 5. 

8. Plot 3;f- j = 8, or 7= 3*- 8. 

9. Show the graphical representation of two num- 
bers whose sum is always 10 ; i . e. of 3 -f y = 10. 

10. Show the graphical representation of two num- 
bers whose difference is always 8. 

11. Graph Zx +y = 12. 

12. VlotS=x-2y. 

13. If you wanted to graph the equation 2x+ky 
= 12, would you tabulate it in this form, or 
would you rewrite it for greater convenience in 
tabulating ? 



Solving Equations with Two Unknowns 237 

14. Graph \x — \y — 8. 

15. Graph 3x = y — 2. 

Section 113. An easier method of plotting a line. A 

straight line is definitely determined or located if any two 
of its points are known. If these points are not too close 
together, they fix the plotted position of the line just as 
accurately as eight or ten points. Therefore, in plotting a 
straight line, it is sufficient to plot only two points, unless 
they are quite close together. 

The easiest points to plot are those on the axes ; that is, 
the points where the line cuts the ;r-axis and the j-axis. 
By referring to Fig. 130, or to the graph of any line, you 
will see that the x-distance of the point in which the line 
cuts the vertical, or y-axis, is always 0, and that the ^-dis- 
tance of the point in which the line cuts the horizontal, or 
x-axis, is always 0. Thus, if we let x be in any equa- 
tion, such as 2x — y = 6, we find the point in which the 
line cuts the jj/-axis. If jt is in 2x — y = 6, we see that 
y equals — 6, which shows that the line cuts the y-axis at a 
point (0, —6); that is, 6 units below the origin. In the 
same way, if we let y be in any equation, we find the 
point in which the line cuts the .r-axis. In this particular 
equation, 2x—y = 6, if y is 0, then x is 3, which shows 
the point in which the line 2x — y— 6 cuts the ^-axis. 

This shorter method requires only the following brief 
table : 

Table 20 



X equals 


O 


? 


y equals 


? 






238 Fundamentals of High School Mathematics 

EXERCISE 94 

1. If x=0, what is y in the equation 2x+y = %} 
What is x if y == ? From these two sets of 
values for x and y, plot the equation. 

2. Given 4:x — 2y = 8. 1 Plot by finding where the 
line cuts the axes. 

3. Where does the graph of 5x-\-2y — 10 cut the 
;r-axis ? thej-axis? 

4. Where does the graph of 2x— Sy = — 6 cut 
the ^r-axis ? the y-a.xis ? Plot. 

5. Graph 2\ x +y = 5. 

6. Plot 3x+2y = 12. 



HOW TO SOLVE GRAPHICALLY EQUATIONS WITH TWO 

UNKNOWNS 

Section 114. When is an equation with two unknowns 
solved? In equations with only one unknown, such as 
Sx + 4 = 19, we found that there was only one value for x 
which would satisfy the equation; namely, x = 5. If we 
substitute 5 for x in this equation, giving 15 + 4 = 19, we 
find that 5 satisfies the equation. Any other number sub- 
stituted for x would not " satisfy the equation." 

But now consider an equation which has two unknowns, 

such as 

■x+y=%. 

Here we see that x might be 3 and y would be 5 ; or x 
might be 6 and y would be 2 ; or x might be 10 and y 
would be — 2. Thus, there are a great many sets of 
values of x and y which could satisfy the equation 
x -\-y = 8. This will be made clear as you work the fol- 
lowing examples. 



Solving Equations with Two Unknowns 239 

EXERCISE 95 

1. Give four sets of values of x and y that will 
satisfy the equation x—y = 6. 

2. Will x = 4 J and y = 3 satisfy the equation 
4 x — y = 15 ? Does x = 5 and r = 4 satisfy it ? 

3. If the equation x-\-y = S is plotted, would the 
points (5, 3) lie on the line representing the 
equation ? (3, 4) ? (10, - 2) ? (1, 7) ? 

4. Does the graph of the equation 2x— y — 7 pass 
through the point (5, 3) ? (4, 2) ? 

We have shown that an equation with two unknowns 
can be satisfied by a very large number of pairs of values 
of the unknowns. Each pair of values that satisfies the 
equation is called a solution of the equation. Therefore, 
"solving the equation " means finding a pair of values that 
will satisfy the equation. 

Section 115. Linear equations. The fact that the graph 
of an equation which contains two unknowns, each of the 
first degree {i.e. no squares or cubes), is always a 
straight line, has led to the name linear equations. Thus, 
2x + y = 5, x -h 5 = 10, etc., are linear equations. 

TWO LINEAR EQUATIONS MAY BE EASILY SOLVED BY 
PLOTTING THEM ON THE SAME AXES 

Section 116. It is a very common problem in mathe- 
matics to have to find one set of values which will satisfy 
each of two equations having two unknowns. For ex- 
ample, what single set of values will satisfy each of these 
equations ? 

ix+y = 8 

\2x-y = 7 



240 Fundamentals of High School Mathematics 



It is clear that x = 6 and y — 2 or (6, 2) will satisfy the 
first equation, but not the second one; in the same way 
x — ^. and y = 4 or (4, 4) satisfies the first equation, but 
not the second one ; x = 6 and y — 5 satisfies the second 
equation, but not the first one. 

OUR PROBLEM IS TO FIND ONE SET OF VALUES THAT 
WILL SATISFY BOTH EQUATIONS 

This can be done, graphically, by plotting both equations 
on the same axes, because in that way we can find a point 
common to the two lines ; that is, the point in which two 
lines intersect. The coordinates of this point will satisfy 
both equations. Figure 
131, on the following 
page, shows both equations 
plotted on the same axes. 
Note that the two lines 
intersect at the point (5, 3). 
This point of intersection 
of the two lines gives a 
single set of values, x = 5 
and 7 = 3, which satisfies 
both equations. (Show that 
x = 5 and y=& checks for 
each of the equations.) fig. 131 

EXERCISE 96 

Find a set of values that will satisfy each of the follow- 
ing pairs of equations, by finding the intersection point of 
their graphs. Check by substituting in the equations. 

x + y = 6 2 iy = 2x+3 

2x —y = 3 \yz=x+ 7 



S K 


5 AJ. 


\ >%t 


S> JL 


s^Sl 


-»^r,y r 


s^ 


ys, 


f s^ 


~/ ^ 


-**- -/ s^s 


1 5p* 


_/ ± %t± 


1 s^ 


i/ -£- s^ 


t s 




i 


_/ 





Solving Equations with Two Unknowns 241 



3. 
4. 
5. 

6. 

7. 

f 

i 



x — y — 5 
2 x +y=l 
2x + y= 12 
x - 2y = - 4 

j/ = ;tr + 6 

a 4- £ = 7 

^ _2£ = -5 

x — y = 5 

j = \ x + 5 



9. 



10. 



11. 



12. 



13. 



\ x — y — 



2x+y = l 

y=x + 10 
x-y=-10 

I2x+y = -ll 

[y — x=l 
U-2y = 2 

y =%x 

y = X-S 

2x-y = ll 



14. { 



[<z + £ = 2 
- 2 5 = 14 



+Y 



-X- 











_^ 




^ 




^ 




5 




^ 




^ 




Si s 




s s 




S v 




s_ s r 




^ s 


v 


r_ S_ 


-S^ _ — -+y 


L S_a 


_ s ^ +A 


*i - 


- S>E 


s . 


St^ 


w 


S*£ 


S: 


t= V* 


s 


^ S^ 




5lv S^ 




£5v V 




- S3^t 




^r- 



Fig. 132 

Section 117. Equations whose graphs are parallel lines, 
i.e. inconsistent equations. Figure 132 shows the graphs of 
the equations represented below. 

(1) f*+>«4 

(2) }2;r+2j=-6 



242 Fundamentals of High School Mathematics 



Note that the lines do not intersect, but are parallel. What 
single set of values of x and y will satisfy each of these 
equations ? Evidently there is none, for they have no point 
in common. Such equations are generally called incon- 
sistent, to distinguish them from the kind that are satisfied 
by some set of values. The latter kind, those whose 
graphs intersect, are often called SIMULTANEOUS EQUA- 
TIONS. 

EXERCISE 97 

Graph each of the following pairs of equations to 
determine which pairs are inconsistent and which are 
simultaneous : 



l. 



y — x = 4 
x — 6 = y 

2x- 



[2y 



4. 



5. 



3=y 
10 = 4* 

f^+j = 4 

x — y = 6 

■yj r 4t=x 

x + y=12 

\x-y = 10 

[2x + y=2 



7. 



10. 



y + "A = x 

fr=_T-3 
12^ = 2*4-12 
\2x-\=y 
I By + 12 = 6x 

lx-S=y 

U-_r = 4 

\x— 6 — y 

y = 10-* 



SUMMARY OF CHAPTER XI 

In this chapter we have learned : 

1. How to locate points on maps and drawings by 
plotting them at given distances from a vertical 
and a horizontal reference line. These reference, 
or base lines, intersect in a point called the origin, 
from which we always plot and record our dis- 



Solving Equations with Two Unknowns 243 

tances. The X-distance on the horizontal axis is 
positive when plotted to the right of the vertical 
axis, and negative when plotted to the left ; sim- 
ilarly the F-distance is positive above the hori- 
zontal axis and negative below it. 

2. The main purpose was to show how to draw the 
picture or graph of an equation by plotting a series 
of points which will satisfy the x and y values of 
the equation. 

3. To plot a straight line only 2 points need be located. 

4. The easiest points to locate are those in the axes. 
For these either x or y is 0. 

5. Two equations may be solved graphically by plot- 
ting the two lines. The values of x and y can be 
found by reading the x and y distances of this point. 

REVIEW EXERCISE 98 

1. From the sum of — 6 and + 10 take — 8. 

2. The product of two numbers is — 40 y; one of 
them is + 10. What is the other ? 

3. State the four principles, or axioms, used in solv- 
ing equations. Illustrate in solving the equation 
by- 8 =+ 2y - 50. 

4. If A = ±x + Sy and B = \x - 3 y, what does 
A + B equal ? What does A — B equal ? 

5. Does - = - — - ? Does - = — ? Does - = — ? 

7 7-2 b be 2 10 

State the principle involved in these examples. 



244 Fundamentals of High School Mathematics 



Age 


Boys who leave school at the 
age of 14 earn weekly wages 
as indicated 


Boys who leave school at the 
age of 18 earn weekly wages 
as indicated 


14 


$ 4.00 





16 


5.00 





18 


7.00 


$10.00 


20 


9.50 


15.00 


22 


11.00 


20.00 


24 


12.00 


24.00 


25 


13.00 


30.00 



6. Studies have been made to determine the money- 
value of a high school education. The table 
above shows the average weekly earnings for 
boys who leave school at the age of 14, and for 
those who remain in school until they are 18 
years old. 

Graph the earnings for each class of boys on the 

same axes. Measure age along the horizontal 

axis. 

Interpret the graph. If a boy knew that he 

would live to be only 25 years old, would it pay 

him, in dollars, to go to high school ? How 

much ? 

7. In solving a particular problem, how do you tell 
whether to use the cosine, or tangent ? Illustrate 
by specific examples. 



CHAPTER XII 

HOW TO SOLVE EQUATIONS WITH TW T UNKNOWNS 
BY ALGEBRAIC METHODS 

SOLUTION BY ELIMINATING ONE UNKNOWN 

Section 118. The need for a shorter method of solving 
equations with two unknowns. In the previous chapter we 
saw that equations with two unknowns can be solved by- 
graphic methods. The exclusive use of that method, how- 
ever, would require a great deal of time, and would 
necessitate that we have cross-section paper at all times. 
Fortunately, there is a shorter method which can be used. 
This is a method by which we eliminate one of the 
unknowns. By eliminating or getting rid of one of the 
unknowns, we obtain an equation with only one unknown. 
The following illustrative examples will explain the differ- 
ent ways by which one of the unknowns is eliminated. 
This chapter will show two methods of elimination. 

I. ELIMINATION OF ONE UNKNOWN BY ADDING OR BY 
SUBTRACTING THE MEMBERS OF THE EQUATIONS 

Section 119. A great many equations in two unknowns 
can be most easily solved by this method. The following 
example illustrates it. 

Illustrative example. Find the value of x and y in these 

equations : 

[2x-y = 5, (1) 

1 x + t/ = 13. (2) 

Adding equation (1) and equation (2) gives 

3x = 18, (3) 

or x = 6. (4) 

Substituting 6 for x in (1) and (2) gives 

y = 7. (5) 

245 



246 Fundamentals of High School Mathematics 



Check: Substituting 6 for x and 7 for y in (1) and (2) gives 

12 - 7 = 5. (6) 

6 + 7 = 13. (7) 

It happens in this example that one of the unknowns, 
y, is eliminated by adding the corresponding members of 
the given equations. In many examples it is possible 
to eliminate one of the unknowns by subtracting the mem- 
bers of one equation from the corresponding members of 
the other. In many other examples, however, it is im- 
possible to eliminate one of the unknowns directly, either 
by adding or by subtracting the members of the two 
equations. For illustration, take this set of equations : 

Illustrative example. 



x-2y 
%x + y 



(1) 
(2) 



If we add the corresponding members of the two equa- 
tions, we get the equation %x — y = 14. But this does not 
eliminate either of the unknowns. In the same way, if 
we subtract (2) from (1), we get the equation — x — Sy = 2. 
Again, this does not eliminate either one of the unknowns. 
This shows that addition or subtraction of the members 
to the equations will not eliminate one of the unknowns 
unless one of them, the x or the y, has the same coefficient 
in both equations. 

Now let us make y in the second equation have the same coeffi- 
cient as y in the first equation. To do so, the second equation 
must be multiplied through by 2. This gives 

4x + 2y = 12, (3) 

*-2y = 8. (1) 



Now, by adding (3) and (1), we get rid of y, obtaining: 



or 



5x = 2( 
x = 4. 



(4) 
(5) 



Solving Equations by Algebraic Methods 247 



Substituting in (1) or (2), 


y = -2. 


(6) 


Check : 


16-4 = 12. 


(7) 




4 + 4=8. 


(8) 



An important question naturally arises here : When do 
we eliminate by addition and when by subtraction ? This 
can be answered by referring to an example. 

*+>=H, (1) 

2^+^ = 4. (2) 

Would either x or y be eliminated by adding the corre- 
sponding members of these equations ? Certainly not, for 
that would give Sx + 2y = 15. Now, would either x or y 
be eliminated by subtracting the members of one equation 
from those of the other ? Yes, for we should have — x = 7. 
However, if the second equation (2) above had been 
Zx — y = 4, then we should eliminate y by adding (1) and (2). 
From these examples we come to the following conclu- 
sions about eliminating one of the variables : 

I. If the variable we wish to eliminate has the 
same sign in both equations, then it is elim- 
inated by subtracting the members of one equa- 
tion from the members of the other equation. 
II. If the variable we wish to eliminate has dif- 
ferent signs in the two equations, it is elim- 
inated by adding the corresponding members 
of the equations. 
III. No variable can be eliminated either by addi- 
tion or by subtraction unless it has the same co- 
efficient in both equations. If it does not have 
the same coefficient in both equations, then we 
must multiply the sides of one, or both, of the 
equations by such a number, or numbers, as 



248 Fundamentals of High School Mathematics 

will make that variable have the same coeffi- 
cient. Thus, to eliminate x in the following 
equations : 



J2x- y 
\3x + 4y 



23. 



(1) 
(2) 



It is necessary to multiply (1) by 3 and to multiply 
(2) by 2. This gives the following equations : 

/6x-3i/ = 24, (3) 

\ 6 x + 8 y = 46. (4) 

Now the variable x can be eliminated by subtracting 
(4) from (3), which gives 

-111/ = -22, 
or y = 2. 



EXERCISE 99 
ELIMINATION BY ADDITION OR SUBTRACTION 

Solve and check each of the following : 



1. 



2. 



3. 



10. 



11. 



f x +j==5 
\x-y = 2 

\2x+3jr=U 

3x — Sj/=- 1 

2r+s=9 
r-s = 12 

X +y=z — 4: 



7. 



8. 



f 3 a _ b = - 2 

±a + £ = -12 
2^-^ = 11 
;tr_3j/=13 
f3J+2<r = 5 

[4^-3^ = 8 



Find two numbers whose sum is 100 and whose 

difference is 18. 

In an election of 642 votes an amendment was 

carried by a majority of 60 votes. How many 

voted yes and how many no ? 

The admission to a school play was 25 cents for 

adults and 15 cents for children. The proceeds 



Solving Equations by Algebraic Methods 249 

from 267 tickets were $ 50.05. How many tickets 
of each kind were sold ? 

12. A purse containing 18 coins, dimes and half 
dollars, amounts to 16.20. Find the number 
of each denomination. 

V+4*~10 



2^ + 3^ = 11 

2^ + 5i/ = 12 
[/ — o ;r= — 1 
f*+2y = ll 

16 - {5* -8 = 8, 

In the remaining examples of this exercise, use any 
method of elimination. 

21. (' = * + '* 25. I'^-zy- 1 



il- 


— J/ = 


-2 


■8> = 


10 




-8 = 


-^ 


+ 4£: 


= 7 


{V. 


+ 3 = 


-4j 


|J = 


11 




+ t>" 


= 12 


-ir 


= -8 



2^+^ = 16 * [ y = ;r + 14.5 



*-^ = 10 [2^-3^ = 7 

2 ^-i^=9 11^-^ + 1 = 

\x=y + 2 27 ' Uj + ^=-15 

4^ = 3j/ + 3 f2;r + 2i/=0 

24. { . ,/ 28. 



by = Qx \x=y + 12 



II. ELIMINATION BY SUBSTITUTION; THAT IS, BY SUB- 
STITUTING THE VALUE OF X FROM ONE EQUATION 
IN THE OTHER EQUATION 

Section 120. This method of elimination will be illus- 
trated by working two easy examples. 



250 Fundamentals of High School Mathematics 



Illustrative example. Find the value of x and of y in 

the following equations : 

>.+ Z/-10, (1) 

x-3z/ = -6. (2) 

Solving equation (1) for x, we get 

x = 10 - y. (3) 

Substituting 10 — y for x in (2) gives 

10-2/-3i/ = -6, (4) 

or -4y=-16, (5) 

or y = 4. (6) 

Substituting 4 for 1/ in (1) or (2) gives 

x = 6. (7) 

Here, as when we eliminate one unknown by adding or 
subtracting equations, our real aim is to get an equation 
which contains only one unknown. We found from equa- 
tion (1) that x = 10 — y. This value of x must be true for 
both equations. (Recall that x is the same for both equa- 
tions, or for both lines, at their point of intersection.) For 
this reason we may substitute 10 — y in place of x in the 
second equation. This gives an equation in one unknown ; 
namely, y. 

The same results could have been obtained by finding the 
value of y, instead of the value of x, from one of the equations 
and substituting it in the other equation. For example : 

Second illustration of the method of eliminating by sub- 
stitution. 

/*+ y = io, (i) 

\x-Zy=-Q. (2) 

From equation (1), y = 10 — x. (3) 

Substituting 10 — x for y in (2) 

x-3(10-x) = -6, (4) 

or X — Z0 + Zx=-Q, (5) 

or 4x = 24, . (6) 

or x = 6, (7) 

and y — 4, as before. 



\y = x 


\ o,r + 4j'= T 


f^+T=8 


i2.r-j' = 10 


' a - 2 /; = - 13 


\ b - « = 9 



Solving Equations by Algebraic Methods 251 

EXERCISE 100 

Solve by the method of substitution and check each 
result : 

l + x — j' = — 15 
(5r-4j = 18 

3. < n 6. 

7. The sum of two numbers is 102 ; the greater 
exceeds the smaller by 6. Find the numbers. 

8. The difference between two numbers is 14, and 
their sum is Q6. Find the numbers. 

9. 12 coins, nickels and dimes, amount to 61.05. 
Find the number of each kind of coin. 

10. The perimeter of a rectangle is 158 inches ; the 
length is 4 feet more than twice the width. 
Find the dimensions of the rectangle. 

11. Bacon costs 10 cents per pound more than 
steak. Find the cost per pound of each if 
4 pounds of bacon and 7 pounds of steak to- 
gether cost 63.48. 

12. A part of 64000 is invested at 4% and the re- 
mainder at 5%. The annual income on both 
investments is 6185. Find the amount of each 
investment. 

13. The quotient of two numbers is 2, and the 
larger exceeds the smaller by T. Find the 
numbers. 

14. Oranges cost 20 cents per dozen more than 
apples. A customer bought 10 dozen oranges 



252 Fundamentals of High School Mathematics 

and 4 dozen apples and received 20 cents in 
change from a 5-dollar bill. Find the price per 
dozen of each. 



15. 



16. 



f 2 x +-3y = 5 

\ 4:x — y — 3 



17. 



_ 3 



= 9 



2~3 
2£_3j 
3 4 



13. 






2.r=3 



EXERCISE 101 
Solve by either method and check each of the following 

fj-3/ = : -'8 



1. 



2. 



7. 



8. 



9. 



5. 



4^+^=14 

2^ + 3^ = 6 

£ = 5^ + 16 

x + 5 y = 1 

[2x+6y = -2 



10. 



11. 



x = 2y-S 
yx— 5y — 21 

x — y = 10 
x=16-2y 

y + 2x=12 

[5x + y = ±2 

The sum of two numbers is 14; the larger ex- 
ceeds the smaller by 2. Find each number. 
Twenty coins, dimes and nickels, have a value 
of $1.70. Find the number of each. 
A boy earns $2 per day more than his sister; 
the boy worked 8 days and the girl worked 6 
days. Both together earned $44. What did 
each earn per day ? 

y=%x-5 
y + x=10 

'2x+3y = 5 
Sx-y = 2 ~" \x=2y-U 



12. 



13. 



2 

3 



x = u-y 

[y = 2x-10 



Solving Equations by Algebraic Methods 253 



14. Three tons of hard coal and two tons of soft coal 
cost $42. At the same prices, 2 tons of hard 
coal and 3 tons of soft coal would cost $38. 
What is the price per ton of each kind of coal ? 

15. The base of one rectangle is 8, and the base 
of a smaller rectangle is 6. The sum of their 
areas is 134, and the difference between their 
areas is 26. Find the height of each rectangle. 

16. A man bought a farm. If he had paid $15 
less per acre, he could have bought 10 acres 
more; at $25 more per acre, he could have 
bought 10 acres less. How many acres did he 
buy, and what did he pay per acre ? 
The difference between two numbers is 6 ; one 
half of the smaller number equals one third 
of the larger. What are the numbers ? 
If oats are worth 50 ^ a bushel, and corn 75 ^, 
how many bushels of each would you have to 
use to make a mixture of 80 bushels worth 
60 ^ a bushel ? 

A man has $10,000 invested. Part of the 
money earns 6 % interest, and the remainder 
earns 5 %. Find the amount invested at each 
rate, if the total yearly income is $ 570. 

[J(^ + 10)-7 = -4 22 - 
-10 



17. 



18 



19. 



20. 



21. 



3 2C? + 2) 

4 4 



x + 



+ 



y 



= 4 



>-5 = £ 

3 2 



-6 



23. 



\Hx-y) 



(x-y) = 26 

2(x + j,)=- 



22 



254 Fundamentals of High School Mathematics 

STANDARDIZED PRACTICE EXERCISE D 

Practice on these examples until you can reach the 
standard, 6 right in 8 minutes. Record your score on your 
record card. 



1. 


(3^ + 7 = 14 

I 6x — 3j/ = 3 . . . . 


7. 


1 2a + 6 = 12 

\5a-3£=19 


2. 


|2/ = 3^+7 

J5/ + 4*=29 


8. 


J7^ + 3j = 31 


3. 


] 2 8 


9. 


U + ^=-S 

{2 5 




[2* = 3j + 15 .. . 


. 


'6/ = bt + 26 


4. 


'\Sd-2c = 8 


10. 


[2x + y = lA 

\bx-2y=2§ 


5. 


|5.r=3j/ + 27 
\3^ + 2jj/ = 20 . . . 


11. 


J 4a = 3* + 9 

13^ + 2^ = 11 


6. 


!H'- 3 


12. 


12^ 3 




[4/ = 5j + 13 




[2^=3j/ + 5 



SUMMARY OF CHAPTER XII 

1. We need shorter methods (than the graphing ones) 
of solving equations with two unknowns. 

2. A valuable short method is to eliminate one of the 
unknowns. 

3. We have learned two ways of doing this : first, 
by adding or subtracting the equations; second, 
by substituting the value of x from one equation 
in the other equation. 



Solving Equations by Algebraic Methods 255 



REVIEW EXERCISE 102 

1. If one tablet costs b dollars, what will x tablets 
cost ? 

2. If a books cost b dollars, what will one book 
cost ? c books ? 

3. What is the perimeter of a rectangle whose 
width is a and whose length is b ? the area ? 

4. What is the width of a rectangle whose perime- 
ter is/ and whose length is xl 

5. The area of a triangle is k. Its base is b, 
What is its altitude? 

6. The sum of two numbers is s. If one is d, 
what is the other ? 

Solve each of the following pairs of equations by any 
method of elimination : 

[3* + 2^=8 

(a-2b = -l 
8 {±a-b = 10 

10. The table below shows how much money (to 
the nearest dollar) you would have at the end 
of a certain number of years if you saved 10 
cents a day and deposited it in a bank which 
pays 3 % interest. 



9. 



*£+-£« 7 

6 4 



2x 
3 



y 



At the end 
of (years) 


1 


2 


3 


5 


8 


10 


14 


17 


20 


total amt. 
saved is 


37 


75 


115 


197 


330 


425 


635 


809 


999 



Represent this graphically, measuring the time 
on the horizontal axis. Estimate the total sav- 
ings at the end of 4 yr. ; 6 yr. ; 25 yr. 



256 Fundamentals of High School Mathematics 

11. A collection box contained 63 coins, nickels and 
quarters. How many of each kind were there 
if the total amount was $ 8.35 ? 

12. Is it ever possible to get from a graph informa- 
tion which could, not be obtained from the 
table ? Illustrate. 

13. The distance from the base to the top of a hill, 
up a uniform incline of 40°, is 800 ft. What is 
the altitude of the top above the base ? 



CHAPTER XIII 

HOW TO FIND PRODUCTS AND FACTORS 

Section 121. Why you should be able to find products. 

Suppose you wanted to find the area of a rectangle whose 



3X 



Area=69C : 



*-5 



Area= 
JO-K. 



-3X+5 
Fig. 133 



dimensions are 3,r + 5 and 2x. To do so, it would be 
necessary to multiply 3^+5 by 2x, or to find the product 
of these two algebraic expressions. One way to do this 
is to divide the rectangle into smaller rectangles, as indi- 
cated in Fig. 133. This gives two rectangles, the dimen- 
sions of one being 2x by & x, and of the other 2x by 5. 
From what you already have learned about multiplication 
you can see that the areas of these are 6x 2 and 10 x, be- 







— 4X > 


-2-| 


Jl 








X 
10 








ii 










-^ 


4X+2- 


>. 



FIG. 134 
257 



258 Fundamentals of High School Mathematics 

cause Sx times 2x is 6x 2 and 5 times 2x is 10 x. Thus 
the area of the original rectangle is 6x 2 -h 10^. Similarly, 
the area of the rectangle in Fig. 134 is what ? What 
would its area be if the dimensions were 6 a -f- 4 and 5 a ? 

These illustrations are given to make clear the need of 
learning how to find products. Other illustrations might 
have been taken. For example, what is the cost of 
15 b -f 3 articles at 4 b cents each ? How much could you 
earn in 6 y + 4 days at Sy dollars per day ? 

A NEW WAY OF INDICATING MULTIPLICATION 

Section 122. As you progress in your study of mathe- 
matics, you will find that it has a language which tells more 
in fewer words or symbols than any other language. For 
example, instead of writing "find the product of Zx+5 
and 2x" it has been agreed to express this by means of the 
parentheses, ( ). Thus, 2x(3x+5) means "to find the 
'product of 3x+5 and 2x" or, "to multiply 3x+ 5 by 
2 x." It is important to note that there is no sign between 
the 2x and the expression in the parentheses. Similarly, 
to state algebraically the problem in the second illustra- 
tion, Fig. 134, you would write" 3x(4x + 2), putting 
no sign between the 3x and the parentheses. Thus, 
5 &(3 b + 7) means to multiply each of the numbers in the 
parentheses by 5 b. 

ORAL EXERCISE 103 
PRACTICE IN FINDING PRODUCTS 

In the following examples, multiply each term within the 
parentheses by the number which immediately precedes 
the parenthesis, or, remove parentheses. 

Illustrative example. 3 x(5 x* + 7 x + 8) = 15 x* + 21 x 2 + 24 x. 



How to Find Products and Factors 



259 



1. 4j(3j + 9) 

2. 6£(2£ + l) 

3. 7 c (3 + 5 <:) 

4. 5 ,r (r — 4 ) 

5. 9(2**+ 7*- 4) 

6. 4tf(fl 2 +3tf + 7) 

7 . 6^(3/ -5^ + 2) 

8. 1(2^ + 3) 

9. 5/(6 — /) 



10. Sy(y-S) 

11. 3r(r+3) 

12. 7 &(!-&) 

13. 8(^ 2 -8^ + 12) 

14. 6(2#-3£+<:) 

15. ad (a + 3 + 1) 

16. *7 2 (x + y + xy) 

17. —1(4 — 57) 

18. -3*(6*-4) 



19. What algebraic expression will represent the 
area of a rectangle whose length is 10 inches 
more than its width ? 

20. What algebraic expression will represent the 
total daily earning of 4 men and 7 boys, if each 
man earns 82 per day more than each boy? 

21. -6 (2* -7) 24. (16/ -7) 

22. — (10— x) 25. (a + b) 

23. -4/(2 y -3) 26. -(b-c) 

In this exercise you have learned how parentheses are 
used to indicate that each of the terms in an expression 
must be multiplied by another number. 

Section 123. More difficult multiplication. Most prod- 
ucts which you will need to find are more difficult than 
those of the preceding exercise. For example : How 
many square feet of floor area in a dining room whose 
dimensions are 4 x + 3 ft. and 5 x + 4 f t. ? To find the 
product of these factors requires something you have not 
yet learned. You know how to multiply 5x + 4 by 4 x or 
Ix + 3 by 5x, but you have not learned how to multiply 



260 Fundamentals of High School Mathematics 

5 % : a-U- 4- - 



T 

10 



Area m 15 x 



Area = 20 X 2 



Area=12 



Area ■ 
16 x 



5X+4 ■ 

Fig. 135 

such expressions as 4jr + 3 by 5^ + 4. The drawing 
shows one way to do this ; namely, the geometrical method 
of dividing the entire area into smaller rectangles, the 
area of each of which you can find. This method gives 
four rectangles whose areas we can find. Thus, we get 
four rectangles whose areas are 20 ;r 2 , 16 x, 15 x, and 12, or, 
collecting terms, an entire area of 20 x 2 + 31 jt + 12. 

Another method of finding the product of 4 x 4- 3 and 
5x + 4 (which is generally written as (4;r + 3)(5^r 4- 4)) 
makes no reference to rectangles. It is very much like 
the method of multiplication Used in arithmetic. To illus- 
trate, this same example could be solved as follows : 

First illustrative example. 

(4x + 3)(5jc + 4). 

4x + 3 
5x4-4 



20 x 2 4- 15 x 

+ 16x4-12 
20x 2 4-31x+ 12 



How to Find Products and Factors 



261 



The 20 x 2 -h 15 x is the result of multiplying 
Alx + 3 by ox, and the 16 x 4- 12 is the result of 
multiplying 4 x -f- 3 by 4. This latter method is 
much more generally used than the geometrical 
method. Let us take another illustration. 

Second illustrative example. 

(7 a + 3) (4 a -5). 
7a + 3 
4a-5 



28 a 2 + 12 a 
- 35 a 



15 



28 a 2 - 23 a - 15 



Note that 7 # + 3 was first multiplied by 4 <?, giving 
28 a 2 + 12 tf. Then 7 # + 3 was multiplied by - 5, 
giving — 35 « — 15. How was the final product ob- 
tained ? 

Which of these two methods, do you think, should 
be learned ? 



EXERCISE 104 




PRACTICE IN FINDING PRODUCTS 


Illustrative example. 




(2x 2 -f 3x+4)(3x — 


7). 


2x 2 +3x +4 




3x-7 




6x3+ 9x 2 +12x 




-14x 2 - 21 x 


-28 



6x 3 

1. (3tf + 2)(4tf + 5) 

2. (7J + 5)(£ + 6) 

3. 0,+ 4)(5j + 3) 

4. (7_5)(2/+8) 



5x 2 - 9x-28 

5. (8j/ + 5)(7j/ + 3) 

6. (6^ + 4)(4^ + 6) 

7. (4a-3)(4«-8) 

8. (x+3)(x-9) 



262 Fundamentals of High School Mathematics 

9. (3*:-4)(3£-9) is. (Jc + 5)(lc-5) 

10. (6*-7)(3*-9) 16. ( a +l)(a-l) 

11. (j/ + 6)(7+6) 17. (3*-2)(3*-2) 

12. (3£ 2 +l)(2£ 2 + 5) is. (^+2^ + 1)^-3) 

13. (4^r 3 +3)(3^ 3 + 10) 19. (j/2 + 67 + 9)(j + 3) 

14. (5a+3)(5a -3) 20. (.r 3 + 5)(> 3 -r- 5) 

21. If you should multiply x+1 by x + l$> what 
would be the first term of your product ? What 
would be the last term ? 

22. Can you tell, at a glance, the first and /<?.$•/ 
terms of the product which you would obtain 
by multiplying 2.r + 7 by 5^r + 4? 

A SHORTER METHOD OF MULTIPLYING 

Section 124. There is a much shorter method of finding 
products like those in Exercise 104. Mastery of this short 
cut will not only save a great deal of time, but it will help 
you in the later work of this chapter. For an illustration, 
take the example 

(3£ + 5)(2£ + 7). 

You have no difficulty in seeing that the first term of the 
product is 6 b 2 {i.e. 3 b x 2 b) and that the last term is 
+ 35 {i.e. 5 x 7). So if there were some method by which 
you could tell the middle term, you could write the product 
at once, without using the longer method of placing one 
factor under the other and multiplying in the regular way. 
To make the new. method clear, it is necessary to refer 
again to the regular way. Let us illustrate with the 
example : 

Illustrative example. 

(3&+5)(2& + 7). 



How to Find Products and Factors 



263 



By the old method : 



26 



6 6 2 + 10 b 

+ 21 b + 35 
6 b" + 31 b + 35 

The arrows show the cross-multiplications or cross-products 
that make up the middle term. One " cross-product " is 2 b times 
+ 5, or + 10 b, and the other cross-product is + 7 times +3 6, 
or + 21 b. Combining the cross-products, we get the middle 
term, + 31 b. 

By the shorter method we get 
+ 21& 
(3&+5)(2& + 7) = 6 ft 2 + 31 b + 35. 
+ 10& 

The curved lines indicate the cross-multiplication, or 
cross-products, which must be combined to give the middle 
term, -f 21 b and + 10 b, giving +31 b. Thus, in this 
new method, it is assumed that you can tell, at a glance, 
the first and last terms of the product. Then you can 
get the middle term by finding the sum of the cross-prod- 
ucts. The curved lines are drawn to help you see the 
cross-products. 

Second illustrative example. 

(4 6+3)(7 6-8). 



24 



The Long Method 


The Short Method 


4 *v^ 




-32& 


28 b 2 + 21 b 


(4 b + 3) (7 b - 8) = 28 b 2 - 11 b 
+ 2f& 


-32&- 


-24 




28 62_ 11 b- 


-24 





It is important to recognize that the curved lines indicate 
cross-products in just the same way that the arrows in the 



264 Fundamentals of High School Mathematics 

long method refer to cross-products. The new method, 
which from now on we shall call the cross-product 
method, enables you to do mentally in much less time 
what was written down by the old method. To give you 
practice in this important method of finding the product of 
two factors the following exercise has been included. 



EXERCISE 105 



Find the products of the following factors by the cross- 
product method : 



1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 



x -I- 2)(x 4- 5) 20. 

2 J + 4)(3 7 4-5) 21. 

£ + 6)(2# + 7) 22. 

X+4:)(3x + 5) 23. 

/-8)(*-3) 24. 

S a + S)(d + 1) 25. 

9;r + 2)(2.r+l) 26. 

4^ + 3)(7r+10) 27. 

5j + 3)(5j-3) 28. 

a + 9)(* + 9) 29. 

s + 2)0 - 5) 30. 

ab + 6)(ab + 3) 31. 

x 2 + 3)(> 2 + 6) 32. 

abc + %){abc - 10) 33. 

2x+3<y)(2x+3y) 34. 

a 4- b)(a 4- £) 35. 

c + d)(c + d) 36. 

/+ ,)(/+*) 37. 

x 4- 5)(;tr — 5) 38. 



.r + 10)(^r-10) 
/ + 4)(j/-4) 
3/ + 7)(3/-T) 
4a + 3)(4a + 3) 
7b + 9y)(7b + 9y) 
j + 3)(j/ + 5) 
^_8)(.r + 8) 
,4-9)0 + 9) 
,-4)0-4) 
3tf4-2)(2^4-4) 
4j-2)(3j + 5) 
2* + 4)(5;r-9) 
^4-4)(^4-7) 
,-9)0+ 9) 
J+T)(i4-7) 
a - 2)0* - 2) 
5* + l)(a-2) 
2* + l)(*-2) 
4^4-3X37-9) 



How to Find Products and Factors 



265 



39. (/ 2 + 3)(/ 2 + 5) 

40. (;>-9)(/ + 2) 

41. (r 2 -3)(^-5) 

42. (/ + 3X^ + 3) 

43. (£ 3 -ll)(£ 3 -ll) 

44. (^ 2 + 3)(> 2 + 3) 

45. (4j+10)(5 7 -8) 



46. (ad + 2c)(ad + 3c) 

47. (%abc—d)(Zabc+d) 

48. (7.r 2 /+j/)(2A-5j/) 

49. (9^ 2 +3j)(9^ 2 + 37) 

50. (8«--4^)(8tf-7» 

51. (/+*)(/+*) 

52. (f-s)tf-s) 




53. Illustrative example. What expression will repre- 
sent the area of a square if each side is x + 6 inches ? 

Solution : Evidently the area is 
(x + 6) (x + 6), or x 2 + 12x + 36. 
This is usually written, however, 
not as (jc + 6)(x + 6) but as 
(x + 6) 2 . The exponent, 2, shows 
that x + 6 is used twice as a fac- 
tor. Thus, (3 x + 5) (3 x + 5) is 
usually written as (3 x + 5) 2 . In 
the same way, (3 x + 5) (3 x +5) 
(3 x + 5) would be written as 
(3x + 5)3. 



54. 0+5) 2 

55. (3 £+5)2 

56. (2 7 + 7) 2 

57. (4^-3)2 

58. (x+yf 

59. (.r-j) 2 

66. O + £)0 + tfT) 

Section 125. Further practice in translating from alge- 
braic symbols into word statements. You have had some 



— X+6 

Fig. 


136 


60. 


(*■+ 


*)" 


61. 


(*- 


^ 


62. 


(/+*) 2 


63. 


(/" 


tf 


64. 


(2* 


+ 3^)2 


65. 


(4* 


-2)2 



266 Fundamentals of High School Mathematics 

practice in translating from algebraic statements into 
word statements. For example, you translated the alge- 
braic expression " a -f- b " into the word statement " the 
sum of two numbers," and the expression " xy" into "the 
product of two numbers." It is important to be able to 
translate into word statements some of the examples which 
you did in the last exercise. 

EXERCISE 106 

Write out the word statement which means the same 
thing as each of the following algebraic expressions : 

1. a—b 7. {a + bf 

2. a 2 8. (a-bf 

3. x 3 9. (a + b)(a - b) 

4. (2yf 10. r* + .y 3 

5. c + d 11. (f+s) 2 =f 2 + 2fs + s 2 

6. a 2 + £ 2 12. (f-sf=f 2 -2fs + s' i 

Section 126. An important special product. One kind 
of multiplication occurs so frequently in later work that 
we should make a special study of it here. It results in a 
very common kind of product. It is the kind represented 
by Examples 11 and 12 in Exercise 106 and by Examples 
54 to 65 in Exercise 105. Look back at these again. 

Note that in each example an expression (either the sum 
or the difference of two numbers) has been squared. The 
expression which is squared contains two terms, and is 
generally called a binomial. The product which is ob- 
tained by squaring the two-term expression (binomial) 
always contains three terms and is called a trinomial 
square. In writing out word statements for Examples 11 



How to Find Products and Factors 267 

and 12, Exercise 106, you probably obtained the following 
results : 

The word statement is as follows : 

The square of the sum of any two numbers equals the 
square of the first number, plus twice the product of 
the two numbers, plus the square of the second 
number. 

(f-sf=f*-2fs + s\ 

For No. 12, the word statement is : 

The square of the difference of any two numbers 
equals the square of the first number, minus twice 
the product of the two numbers, plus the square of 
the second number. 

These statements are frequently used as rules for squar- 
ing the sum or difference of two numbers, i.e. for squaring 
binomials. 

EXERCISE 107 

Square the following binomials by the most economical 
method that you know : 

1. (x+yf 9. (2^ + 5) 2 

2. (a + bf 10. (3j-4) 2 

3. (c-df 11. (10tf-2) 2 

4. (x-yf 12. (5c-Sf 

5. {2x+yf 13. (2^ 2 -3 yf 

6 . (j + 10) 2 14. (5^ 2 -4/) 2 



7. O-8) 2 15. O + l) 2 

3 



8. (j 2 -8) 2 16. {y-lf 



268 Fundamentals of High School Mathematics 

17. (2y -J) 2 22. 

18. (6£ + J) 2 ' 23 

19. (5^-i) 2 

20. (9^ 2 --i-) 2 24> 

25. 



0" 


fi/) 2 


(*• 


i)' 


s- 


-)' 


(5* + iJ» 



21. Lr + 

V 26. (3^-i) 2 

27. In squaring each of these binomials, how many- 
terms did you get ? 

28. How is the first term always obtained ? the last 
term ? the middle term ? 

29. If you knew the trinomial square, could you tell 
what binomial had been squared to give it? 
How? 

30. State what binomial has been squared to give 
each of the following trinomial squares : 

(a) x l + i Lxy + y 1 (e) d 2 + 14 df + 49 f 2 

(£)j/ 2 + 6j/ + 9 (/) 25^- 2 + 30^/ + 9j/ 2 

(c) c 2 + 10 c + 25 {g) 16 b 2 -80bc + 100 c 2 

(d) a 2 -12 a + 36 (k) 30 y + 225/ + 1 

31. The following are the first and third terms of 
trinomial squares. Find the middle term for 
each. 



(a) y 2 + ? + x 2 


(0«"-? + A 


(b) c 2 - ? + 25 


(J) 9x 2 + ? + 25y 2 


Or) a 2 + ? + 16 


(k) 4^ 2 +? + l 


(d) 25.r 2 +? + l 


(/) 100 x 2 y 2 + ? + 25 


(e) 16 a 2 + ? + 9 


O) c 2 a 2 + ? + x 2 


(/) 36/+? + 9 


(«) t* ~ ? + * 


Qr)y + ? + i 


W^ 2 -?+2 4 5 


(^)^r 2 +? + l 


0>)/ 2 +? + 49/ 4 



How to Find Products and Factors 



269 



32. The following are the first and middle terms 
of trinomial squares. Find the third term of 
each : 

(a) x 2 + 6x + ? 
(£) f - % y + ? 

(c) c 2 - 10 y + ? 
(d) b 2 +20& + ? 

(e) a 2 P - 12 ab + ? 
(/) x A - 14.r 2 + ? 
(^) f _ ±y> + ? 

(k) ±x 2 + l2x+ ? 
(/') 9j 2 + 3O7 + ? 



(/) 16^ 2 + 40^ + ? 
(yfc) 25^ 2 - 80 r + ? 
(/) -r 2 + * + ? 

(^) J/ 2 _j_ y _j_ ? 

(») a 2 - a + ? 
(<?) b 2 + 1 £ + ? 
(/) t* + \r+} 

(^)^ 2 -I^ + ? 
( r ) 4 / 2 + / + ? 



STANDARDIZED PRACTICE EXERCISE E (TIMED) 

Practice on this exercise until you can reach the standard, 
14 examples right in 4 minutes. 



1. (2x-vZy) 2 10. (4j+7)(3j-2). 

2. (j 2 -5) 2 11. (6a.+ 5bf 

3. (2^ + 3) (2 a- 3). 12. (r 2 -8) 2 

4. (2 £ 3 + 1) (2 £ 3 + 3) 13. (5^+3)(5^-3) . 

5. (3^ + 2) (2*- 5). 14. (4/ + l)(4/+2) 

6. (5^ + 4w) 2 15. (6 ^+2) (2 t-3) . 

7. (/ 3 - 3) 2 16. (2 * + 7<r) 2 

a (4# + 5) (4^-5). 17. 0*-7) 2 

9. (3 ^ + 4) (3^ + 5) 18. (3^ + 4)(3j/-4). 



Section 127. How to solve equations which involve prod- 
ucts. The solution of a great many equations depends 



270 Fundamentals of High School Mathematics 



upon your being able to find products like those you have 
just been finding. To illustrate, consider the problem : 

Illustrative example. The length of a rectangle is 6 inches 
more than, and the width is 2 inches less than, the sides of a 
square; the area of the rectangle exceeds the area of the square 
by 20 square inches. What are the dimensions of each ? 
Solution : Let s = the side of the square. 
Translating into algebra, we have the equation: 

(s + 6) (s - 2) = s 2 + 20. 
Multiplying, or removing parentheses, gives 

s 2 + 4s-12 = s 2 + 20. 
Subtracting s 2 from each side gives 
4 s- 12 = 20. 
.-. s=S. 
Thus, the sides of the rectangle are 14 and 6. 
Check this result. 

EXERCISE 108 

PRACTICE IN SOLVING EQUATIONS WHICH INVOLVE PRODUCTS 

Solve and check each of the following equations : 

1. (x + 6) 2 = ^ 2 +96 

2. (r+3)0/ + 6)-y = 63 

3. (2£ + 3)(£ + 4) = (£ + l)(2£ + 10) 

4. One number is 5 larger than another; the 
square of the larger exceeds the square of the 
smaller by 55. Find each number. 

5. The length of a rectangle is 8 inches more 
than, and its width is 3 inches less than, the 
sides of a square ; the area of the rectangle ex- 
ceeds the area of the square by 26 square inches. 
Find the dimensions of the rectangle. 

6. (7 + 5)(^-5) = (j/-6)( 7 + 2) 
7.' 2(x - 8)- 30 - 4)= - 5x 
8. (£ + 4) 2 -(£-2)2 = 10 



How to Find Products and Factors 

9. (x+ 3)2 ~{ x -lf*= 40 

10. O + 5) 2 -0/ + 4)2 = -l 

11. (^-8) 2 =(,r-12)2 

12. 2(/+3)2 = 2(/-8) 2 

13. 3(.r + 6)(> + 4) = (3.r + 1)(* + 9) 



271 



HOW TO FIND THE FACTORS OF AN ALGEBRAIC 
EXPRESSION 

A. FINDING THE COMMON FACTOR 

Section 128. Meaning of the word FACTOR. If you 
know that the area of a rectangle is 24 square inches, what 
might be its dimensions ? You readily see here that the 
dimensions might be 4 inches and 6 inches ; or 3 inches 
and 8 inches ; or 12 inches and 2 inches, because the 
product of 4 and 6, or of 3 and 8, or of 12 and 2, is in each 
case 24. This process of finding the numbers which, when 
multiplied together, give another, is called FACTORING. 
The numbers you find are called the FACTORS. Thus, 4 
and 6 are factors of 24 ; also 3 and 8, or 12 and 2. 





Area= 5X + 35 


t 

LO 

V 


-e= — 


? ^ 





Fig. 137 



The same reasoning is used in algebra as in arithmetic. 
For example, suppose the area of a rectangle is 5 x -f- 35 
square units. If the width is 5 units, what must the length 



272 Fundamentals of High School Mathematics 

be ? Similarly, if the area is 4^r 2 -f 28 x and the width is 
4 x, what is the length ? 

Section 129. What is a common factor ? Now let us 
take a more difficult illustration. In the previous two 
cases, you knew both the area and one dimension, or the 
product and one of its factors. But in most factoring prob- 
lems you do not know any of the factors. For example, 
what are the dimensions of the rectangle whose area is 
7 x + 21, or, in other words, what are tJie factors of 
1 x +21? A study of the Ix and 21 shows that 7 'is 
a factor of each, or is a COMMON FACTOR of both terms. 
What, then, must 7 be multiplied by to give 7jt+ 21, or, 
what must be the length of this rectangle if its width is 7 ? 
Evidently, it'must be x + 3. This problem should be written 

7* + 21 = 7(x + 3). 
A second illustration should make clear what is meant by 
factoring in algebra. 

Find the factors of ax + ay + aw, or find the dimensions 
of a rectangle if its area is ax + ay + aw. 

By observing each term, we see that a is a factor common 
to all the terms. 

Dividing each term of the expression by a gives the other 
factor, x +y + w. Hence, ax + ay + aw = a(x^-y + w\ 
and a is one factor and x+y + w is the other one. 

These illustrations are intended to make clear how to 
factor an expression in which there is a factor common 
to all the terms. 

EXERCISE 109 

Factor each of the following expressions : 

1. 3;r+12 3. ax+bx 

2. 5 a - 20 4. 7 a - 21 



How to Find Products and Factors 



2 73 



5. 


17,r+34 


17. 


;r 2 + jt 


6. 


6^ + 9 


18. 


a 2 + 20 a* 


7. 


5 + Ua 


19. 


x 2 + 5x* 


8. 


9 + 6* 2 


20. 


# + #£ -f- a 2 


9. 


4j 2 + 12 


21. 


x 2 -bx± 


10. 


8^ + 12£ 


22. 


±a*-12a 5 


11. 


ab + ac + ah 


23. 


5 x 2 y — bxy 


12. 


5 +10 a + 15 a 2 


24. 


7 £ 3 - 21 £ 2 


13. 


2x 2 -±xy + 2j, 2 


25. 


12 ^ 4- 6 « 


14. 


2-8^ 2 


26. 


^ + 2*/ 


15. 


5^ r 2_5y 


27. 


2- 20* 


16. 


4^2 +8 ^ + 4^2 


28. 


64* 2 -21^ 



29. 6 ^* 2 — 12 «* 3 

Can you check the examples in this exercise ? 
this one : 

5x 2 - 15x± = 5*2(l_ Sx). 



Check 



B. THE CROSS-PRODUCT METHOD OF FACTORING 

Section 130. In the previous section all the expressions 
which you factored had a common factor. But most ex- 
pressions which you will need to factor are much more 
difficult than those ; and they do not always contain a 
common factor. 

Illustrative example. Factor 2 x 2 + 5 x + 3, or find the di- 
mensions of a rectangle having this area. 

From what you learned about products, you can see 
that this expression was very likely made by multiplying 
two factors together. Also, you can see that the first 
terms of the two factors must be 2* and x. To help you 
to get the correct result, always write the blank form of 



274 Fundamentals of High School Mathematics 

the two parentheses first, thus: ( )( ). Then as 
you determine each term of each factor, you can write it 
in the appropriate place. Later you will doubtless be 
able to do all the work in your head and not have to 
write out the steps. Second, therefore, write the first 
terms in the blank form, thus : 

2jt 2 + 5x + 3=(2x )(jc ). 

Third, you have to find the second term of each factor. 
From your previous work in finding products, you know 
that the last term of the expression, + 3, was obtained by 
multiplying together the second terms of the two factors. 
Then the second terms must be such that their product is 
+ 3. Obviously, they are 1 and 3 or 3 and 1. To tell 
whether the 3 or the 1 belongs in the first factor we have 
to try it, and test or check to see if the middle term will 
be correct (+ 5x). Trying this out, we have : 

6x 

2jt 2 + 5jc + 3=(2x + l)(> + 3). 



Checking, — that is, multiplying the two factors together, 
— we see that this does not give the correct middle term, 
for + Qx and + x are not 5x. But, we might interchange 
the 1 and 3. Trying this, we get 

2x 
2x 2 + 5 x + 3 =(2x + 3)(x + l). 
3x 

Multiplying these together shows that our factors are 
correct, for their product gives the original expression, 
2x 2 + 6x+ 3. 

Let us try another example : 



How to Find Products and Factors 



275 



Second illustrative example. Factor 5 x 2 - 36 x + 7. 
First write the blank form thus : ( ) ( ) . 
Next we can tell at once that the first terms of our factors are 
5 x and jc, giving 

5x 2 -36;c + 7 = (5x )(jc ). 

Examining the last term of the expression + 7, we see that the 
second terms of the required factors must be 1 and 7 or 7 and 1. 
Trying out the 1 and 7 gives 



5 x 2 - 36 x + 7 = (5 x + 1) (jc + 7) . 

But the check shows that the sum of the cross-products is + 36 x, 
whereas it should be — 36 x. This can be corrected by chang- 
ing the sign of the second terms to — 1 and — 7, giving 

(5 x- l)(x -7), 

The result of checking shows these to be the correct 
factors. 

Section 131. The sum of the cross-products must equal 
the middle term. These two explanations have been given 
to show the importance of getting, as factors, expressions 
such that the sum of the cross-products will give the middle 
term of the expression to be factored. It is assumed that 
you can tell, at a glance, what the first terms might be, 
and what the second terms might be by looking at the 
first and last terms of the expressions which you want to 
factor. 

For example, in factoring 

6^ 2 + 13^+6, 

the first terms might be 3x and 2x, or 6^ and x\ the 
second terms might be 3 and 2, 2 and 3, or 6 and 1. But 
since the product of the factors must give the original 
expression, we can tell by trying these various possible 
combinations that the factors are 



276 Fundamentals of High School Mathematics 



±x 



( 3* + 2)(2;r + 3 ). 

No other arrangement of numbers will give the correct 
middle term, 4- 13 x. 

From this explanation you should be able to factor the 
expressions in Exercise 110. Don't be discouraged if you 
have to try more than once before you succeed. Difficult 
tasks often require many trials. 



EXERCISE 110 
FACTORING BY THE CROSS-PRODUCT METHOD 

Check each example carefully. 

(The parentheses are written here to suggest to you 
how to begin.) 

1. ^ 2 + 5^+6 = ( )( ) 

2. / + 10 7 +21 = ( )( ) 

3. 2 .r 2 + 7 * + 5 = (2* 4- ?)(* + ?) 

4. *a + 6<: + 9 = ( ')( ) 

5. ^2_ 8 ^ + 12 = ( )( ) 

6. 5y 2 + 16j/ + 3 = (5j/+?Xj/+?) 



7. « 2 + 12^ + 36 = 

8. 5 x % + 8 jt + 3 = 

9. ^ 2 -2^r-24 = 

10. m 2 — m — 20 = 

11. 2£ 2 +13£ + 15 = 

12. 3.r 2 - 13* + 4 = 

13. /2_5,_4o == 



14. ioy + 13j/-3 = 

15. 4* 2 + 20*+25 = 

16. a 2 + tf-72 = 

17. y-16 = 

18. £ 2 -25 = 

19. 15^-31^ + 14 = 

20. 21£ 2 -£-2 = 



How to Find Products and Factors 



277 



21. 6x 2 -6x + 1 

22. 3x* + ±x + l 

23. 2j 2 -j/-28 

24. 2a 2 + 1a + S 

25. x*-llx + 2i 

26. j/ 2 - 10j/ 4- 20 



27. tf 2 + 6tf + 9 

28. j/ 2 - 87 + 16 

29. / 2 +10 

30. c 2 + c-30 

31. 3^ 2 + 8^ + 5 

32. 2^ 2 -5.r+3 



Section 132. Importance of finding the prime factors. 

Any algebraic expression that cannot be factored is PRIME. 
For example, 3 x -f 5 is prime, because there are no integral 
expressions which can be multiplied together to produce it. 
But 9 x +6 is not prime because it can be obtained by 
multiplying 3 and x -f 2. 

It is important that you should always find. prime factors. 
To illustrate : 

First illustrative example. Factor 3 & 2 - 21 b +36. 

By inspection, we see that 3 is a common factor. 
Removing it, we have 

3(& 2 -7&+12). 

Now, unless we remember that prime factors should be found, 
we are likely to leave the example in this incomplete form. 
The b 2 — 7 b + 12 can be factored further, however, giving 

(&-3)(&-4). 

Thus, the original example should be factored as follows : 
3 ft 2 - 21 b + 36 = 3(# - 7 b + 12) 

= 3(&_4)(&_3). 



Second illustrative example. Another illustration will 
make clear the importance of finding prime factors. Suppose we 
wish to factor 2 x 2 — 50. As in the previous example, we always 
first look for a common factor. This gives 

2(x 2 -25V 



278 Fundamentals of High School Mathematics 

Now, again, we are apt to leave the example in this incomplete 
form, not remembering to see if we can further factor x 2 — 25. 
We see, however, that we can. The complete solution is : 
2x 2 -50 = 2(x 2 -25) 

= 2(x + 5)(x-5). 

These explanations are given to help you keep in mind 
that in all factoring work there are two absolutely essen- 
tial steps ; namely, 

1. LOOK FOR A COMMON FACTOR. 

2. FIND PRIME FACTORS; I.E. FACTOR COMPLETELY. 

EXERCISE 111 
PRACTICE IN FACTORING COMPLETELY 



1. 


2^ 2 + 14« + 24 


17. 


49^+70^ + 25 


2. 


5/ -45 


18. 


5^_80 


3. 


st 2 -st -20 s 


19. 


x s — X 2 


4. 


1a 2 -Ua-105 


20. 


6^ 2 -18^ 


5. 


3^ 2 + 12^ + 45 


21. 


3 y _ 3j{/ _ 36 


6. 


&-6.X+9 


22. 


12x 2 + 31x-10 


7. 


6/2-15 /3 


23. 


10^ + 25+^2 


8. 


2 - 128 t 2 


24. 


J2 + 10 


9. 


ab 2 -ab-12a 


25. 


^2 + 36 


10. 


6^-2 + 13^+6 


26. 


2^ 2 _8 


11. 


3a 2 + a-2 


27. 


3 £ 3 + 27 


12. 


2a 2 -5a + S 


28. 


3^_12^_180 


13. 


7b 2 -17b _12 


29. 


2^ + 10^-168 


14. 


^2 _ 12 q _ 28 


30. 


^ r 2_ ;r _ll 


15. 


2^ _ 14^24 


31. 


2j2_ J/ _i 


16. 


y_6y-16^ 


32. 


6^2_ 4 ^_2 



How to Find Products and Factors 



279 



33. 3* 2 + 4.r + l 

34. 20;r2 + 70* + 60 

35. 2 £2- £-3 

36. 2tf 2 + 18 

37. 5 y — 15j/ 2 

38. a 2 -b 2 

39. 3^ 2 -^-10 

40. 2*2 + 3*- 9 

41. 6j/ 2 +j— 15 

42. 8*2 _ 2 

43. 18/2-50 

44. 9* 2 + 17*-2 



45. 6 a 2 -0-12 

46. 7j/ 2 - 9^-10 

47. 2* 2 -36* + 64 

48. f - 3 r - 4 

49. / 2 -16 

50. / 2 + 16 

51. 2 j/ 2 - 2j/- 24 

52. £ 2 - 24 + 2 £ 

53. _ 30 _ * + *2 

54. 4 * 2 + 20 * + 25 

55. c* + 2c 2 + C 

56. / 4 - 81 



STANDARDIZED PRACTICE EXERCISE F (TIMED) 

Practice on this exercise until you can reach the 
standard, 12 examples right in 4 minutes. Factor each 
expression. 



1. 6 * 2 - 18 * 3 . . . . 


10. 4/ 4 - 3/ 

11. 2j/-16j 3 

12. 5 * 2 + 15 * + 10 

13. 6 - 24/ 

14. P + 6 £ 3 + 5 £ 2 . . . 

15. 5 r 6 - 8 r 9 

16. 3 a 2 - 15 a 5 ..... 

17. 4 w 2 + 16 zv + 12 . 

18. 7 - 21 q 


2. 2/ + 10j/ + 12. 

3. 5 - 30 d 2 


4. r 4 + 7^3 + i 2r 2 

5. 3 ^ 3 - 11 # ... 

6. 7 * 3 - 14 * 5 ... 

7. 3 £ 2 + 18 b+ 24. 

8. 9 - 18 q 


9. * 5 +3* 4 +2* 3 . 



280 Fundamentals of High School Mathematics 



STANDARDIZED PRACTICE EXERCISE G (TIMED) 

Practice on these examples until you can reach the stand- 
ard, 12 examples right in 6 minutes. Factor each expression. 



1. S x 2 ~6x- 24 . 

2. 9* 4 -25j/ 2 

3. p± - p 2 -20 . . . 

4. p 2 4-5/4-10 . 



10. 25 d 2 + 30 dc + 9 c 2 

11. 5 tf 2 - 20 a - 60 . . 

12. ** - 81 w Q 

is. f -y -so 



5. 


9;tr 2 + YLxy + ±y 2 


14. 


^ 2 + 7 *r 4- li 


6. 


2y2 _ Qy _ 20 . . 


15. 


49/ 2 4-28/^ + 4^ 2 


7. 


16 ^ _ 49^ 


16. 


3 p - 15 b - 150 . 


8. 


a* _ ^-3 _ 12 


17. 


4 ;* 10 - 121 .r 8 


9. 


^ + 3^+ 8 


18. 


jr 2 — ^r — 56 



REVIEW EXERCISE 112 

1. What is the area of a square formed by adding 
4 ft. to the sides of a square x ft. long ? 

2. What does (x — 4)(>4-6) represent, if x repre- 
sents the side of a square ? 

3. A rectangular field 5y rods long has a perimeter 
of 24 y rods. What expression will represent 
the area of the field in square rods ? in acres ? 

4. If the quotient is represented by q, the divisor 
by d, and the remainder by r } what will repre- 
sent the dividend ? 

5. If a park is w rods wide and 1 rod long, how many 
miles would you walk in going around it n times ? 

6. How do you divide a product of several factors 



How to Find Products and Factors 281 

by a number ? For example, in dividing 12-3-6 
by 2, would you divide each factor by 2 ? 

7. How do you multiply a product of several fac- 
tors by a number ? Give an illustration. 

8. The product of four factors is 60. Three of 
them are 2, 3, and 5. Find the fourth factor. 

9. How much do you increase the area of a square 
whose side is x y if you increase its side 4 units ? 

10. Make a detailed summary for this chapter. 

11. Solve for „r, explaining each step : 

-4^+6 = 2^-18. 

12. In what way is factoring like division f How is 
it like multiplication ? 

13. Solve: |6j/-^ = 7H-4 7 , 

14. Make up five examples for the class to factor, 
and then give them to the class to work. 

15. How many terms do you get when you square 
the sum of two numbers, e.g. (2x+3j/f? 
when you square the difference of tzvo numbers, 
e.g. (4 # - 3 bf ? 

16. Evaluate {a + bf if a = — 3 and b = 4- 1. 

17. Does {a + bf = a 2 4- b 2 ? Show by using 4 for a 
and 5 for b. 

18. A tree stands on a bluff on the opposite side of 
a river from the observer. Its foot is at an 
elevation of 45° and its top at 60°. Which has 
the greater height, the bluff or the tree ? What 
measurement would you have to make to find 
the height of the tree ? the width of the river ? 

19. Translate : a 2 - b 2 = (a + b)(a - b). 



CHAPTER XIV 

THE USE OF FRACTIONS WITH LETTERS 

Section 133. What an algebraic fraction means. In 

arithmetic a fraction was used to represent one or more 
of the equal parts of some unit. For example, the frac- 
tion | meant 4 of the 5 equal parts into which something 
had been divided. In algebra, however, a fraction has a 
more general meaning. It is thought of as a quotient, or 

an indicated division. Thus, the fraction - means "the 

b 

quotient of a and 3," and is read "a divided by b" ; or "a 
over b. ,} Examples of algebraic fractions are : 
D*N _3 a + b 

2-5' r ab ' 

Section 134. Numerator and denominator : the terms of 
a fraction. Just as in arithmetic, we have the terms 
numerator and denominator of fractions. The numerator 
is the dividend, and the denominator is the divisor. 

Section 135. How to change fractions into other equiva- 
lent fractions. In arithmetic we frequently change frac- 
tions into equivalents, for example, t 8 q may be changed to 
|, or f may be changed to |- by dividing the numerator 
and denominator of each fraction by 2. In the same way 

— may be reduced to the equivalent, -, by dividing the 
ac c 

numerator and denominator by a. Similarly - — *— '- — — 

J J (a + b)(a-b) 

is reduced to by dividing both numerator and de- 

a — b 

nominator by {a + b). 

EXERCISE 113 — ORAL WORK 

Tell what has been done to the first fraction in each of 
the following examples to give the corresponding equiva- 
lent fraction : 

282 



The Use of Fractions with Letters 



283 



i- if 



o 1 U 
Z - 18 



3. 



5. 



« _ 


ac 
Tc 


^_ 


X 


tfj/ 


a 


x*y 


= *! 


xw 


w 



6. 



8. 



9. 



10. 



jM = 


/ 




rq 


r 




a 2 bc 
~a~b~' 


= ac 




ab 2 
a 2 be 


_b_ 
ac 




2-3 

2-5 


3 
5 




54 


• 2 


4 


7-5 


.2 


7 



11. 



12. - = 



— 13. 



14. 



15. 



a • b • c 


; 


a - c ' d 


</ 


x _ «&r 

y aby 




x+1 


2^ + 2 


4 


8 


j + 2 

5 


3(r+2) 

15 





5tf 



a + £ 5(> + 6) 



TWO IMPORTANT PRINCIPLES IN HANDLING FRACTIONS 

Section 136. The examples in Exercise 113 illustrate 
two very important principles in fractions. 

I. To multiply both numerator and denominator of 
a fraction by the same expression does not change 
the value of the fraction. 
II. To divide both numerator and denominator of a 
fraction by the same expression does not change 
the value of the expression. 
These principles are used in changing fractions into 
equivalent fractions. The next exercise will give practice 
in doing this. State what principle is used in each change. 

EXERCISE 114 
PRACTICE IN CHANGING FRACTIONS INTO EQUIVALENT FRACTIONS 

Supply the missing term in each of the following, and 
tell what fundamental principle has been used in changing 
the first fraction into the second fraction : 

3 ? 2 - = — 

' 3 ? 



1. - = 



284 Fundamentals of High School Mathematics 



ab a 
3. — — — 

be ? 



4. 



x + 5 3x + 15 

4:b ? 

a + 3 5^ + 15 

x+y _ ? 



^r — y ix — y)(x+y) 
a -2 ? 



^ + 6 + 6)(> + 3) 

.r + 3 ? 

(jr + 5)(jr+3) .r + 5 
b 2b 



10. 



11. 



12. 



13. 



14. 



15. 



tf + £ 


? 




a — b a 2. 


-2^ + ^ 2 


x+2 


? 




.r+3 x 2 


+ 5x 


+ 6 


b 


? 




X + 5 ;tr 2 


-1-5^ 




+ £)(<Z - 


-*)- 


^ + ^ 


(*-*) 




? 


(* + 2) 




? 



tf + £ 



17. 



16. 

2 -6^ + 9 



(x + 2)(x-2) x-2 

/> + 5 = 1 

(/+5X/4-2) ? 

a- 2 +7^ + 12 ? 



^ 2 -8^ + 15 rt-5 



Section 137. Reducing fractions to lowest terms. You 
have been taught in arithmetic always to reduce fractions 
to their lowest terms. It is equally important to reduce 
algebraic fractions to lowest terms. Just as in arithmetic, 
a fraction is in its lowest terms when it has no factor 
common to both numerator and denominator. Hence, to 
reduce fractions to lowest terms, it is necessary to divide 
each numerator and denominator by all factors common 
to both. 



EXERCISE 115 
PRACTICE IN REDUCING FRACTIONS TO THEIR LOWEST TERMS 

4 3 14 a be x*y 



1. Reduce to lowest terms : 



20' 15' 36' ab" cd* xy' 



The Use of Fractions with Letters 285 

2. Reduce to lowest terms : — -, — ^-, — — , ^-f^. 

fl^/ b-TW 9<2^ O^J/ 4 

Reduce each of the following to lowest terms : 

4a 3 ^ 2 5 2 . wy*ufi 2(a + b) 

12ad 2 <? ' I2yw± ' 6(a + b) 

±>5-x 120 a 2 bc± 30(x+y) 

' 2-5-y * 2-3.4«V ' 6(> + 2j/) 

Factor numerator and denominator of each of the fol- 
lowing, and then reduce to lowest terms : 

a 2 -b 2 „ 9b 2 -c 2 

9. 13. 

a 2 + 2ab + b 2 

10. • . * 14. 



a 2 


-b 2 


a 2 + i 


lab + b 2 
-4 


b 2 + \ 


r b + 12 


b 2 + 
%a 2 - 


bb + 6 
-30 



11. — r 1 zrr 2 15. 



12. — 16. 

Qx+I2y 



9b*- 


-Qbc + c 2 


t 2 - 


4 


2/ + 4 


(* + 


■2) 2 


x 2 - 


-4 


a 2 - 


-oi 



17. 



ax 



5a 



x 2 -25 
HOW TO ADD OR SUBTRACT ALGEBRAIC FRACTIONS 

Section 138. Use of the most convenient denominator. 

We learned in arithmetic that fractions may not be 
added or subtracted unless they have the same de- 
nominator. Only like fractions may be combined by 
addition or subtraction. Therefore we had to change 
fractions into equivalent fractions which had the same 
denominator. 

For example : to add f and -| we must change each 
fraction to 24ths. 

The same reasoning holds with fractions with letters. 



286 Fundamentals of High School Mathematics 

We must make the fractions have the same denomi- 
nator. 

For example, to add - + - , we must change each fraction into 
b d 

an equivalent fraction in which bd is the denominator, just as 

24 is the denominator in the foregoing example. 

Illustrative example. Showing how to change fractions 
into equivalent fractions which have the same denom- 
inators. 

Change the fractions — and — into equivalent fractions hav- 
& bd cd H 

ing the same denominator. 

The most convenient denominator must contain b, d, and c, and 

be of lowest possible degree. Consequently it is bdc. 

To change the first denominator, bd, to bdc, it is necessary to 

multiply the numerator and denominator by c. (Why multiply 

the numerator also by c ?) This gives — as a fraction equiva- 

bdc 

lent to — ♦ In the same way the second denominator, cd, must 
bd 

be changed to bdc. Multiplying both terms of the fraction — 

cd 

b 2 
by b, gives — as an equivalent fraction. Thus we have changed 
bdc 

the two given fractions into equivalent fractions which have the 

same denominator. 

~ a ac , b b 2 

Summarizing : — = and — = 

bd bdc cd bdc 

Section 139. How to find the most convenient denomi- 
nator. It has been pointed out that fractions must have 
the same denominator before they can be combined. In 
the following exercise you are to decide what the most 
convenient denominator is in each example, and then 
to make the fractions in each example have that denomi- 
nator. 



The Use of Fractions with Letters 



287 



EXERCISE 116 



MAKE THE FRACTIONS IN EACH OF THESE EXAMPLES HAVE THE SAME 
DENOMINATOR 



1. 



3. 



13. 



14. 



a c 

~V ~b 


. ab cd 
c a 


a-\-b a—b 
5 x ' x 


10. *-, 

wy 


wx 


x 3 zv 

/ y 


5 a - x 
b' 4 


b+Z b 

y %y 


*■% 


ab 2 
c*d 


a c 
H d 


2y y 


9. — — 

be ab 


-% 


e* 
lOy 


5 


4 


15 5 


3 




a + b' 


a +b 


• i-2 


b-Z 




X 


X 


1. *+ 2 . 


X+4: 





x+2' x+S 



+ 3 x + b 



17. From the experience you have had in solving the 
previous examples, you will be able to answer the follow- 
ing questions concerning how to find the most convenient 
denominator. 

(a) What is the most convenient denominator for two 

fractions which have the same denominator ? 

(6) What is the most convenient denominator for two 

fractions which have unlike denominators with no 

factors common to each ? Give some illustrations from the 

preceding exercise. 

(c) What is the most convenient denominator for two or 

more fractions which have unlike denominators, but 

which have a factor common to both denominators? 

Illustrate from the preceding exercise. 



18. Add § and f 


20. Add | and |. 


19. Add ^ and f 

b b 


21. Add 2 and -; 

b d 



We have learned how to change fractions into equivalent 
fractions which have the same denominator. Now we are 
ready to get practice in adding and subtracting fractions. 



2&& Fundamentals of High School Mathematics 



EXERCISE 117 
ORAL PRACTICE IN ADDING AND SUBTRACTING FRACTIONS 

§4.5 ?4_? ^,5^ * , £ y_ &y_ 2 3 
2 + 2'5 + 5'2 + 2 '3 + 3'4 + 4 '* "V 



2. 



a_ c , x 
b b' y 



2 2a_a.Sx x, 2y 5y t a + b a — b 

5 ; T"~5 ; 1[ 4'T T ; "~6~ ~~5~" 

^. ^_^. iLi__£. ?/.§_£ 
jj/' <? ^' 3 3' 5 5 

4 2 i 1 . 2_i__3_. A_ 1_ • 6 _L 1 • 1 _ • 1 • 2 i i_ 
rr ' If ~ "T > 5 '10' 6 T4» 4^8> 3 6' 9^3' 

J_b 

4 * 



5. 



^_i_^i!. :t£-4_ ^ • -^ _ ?J^- 3;r 9;r # £ 

2 4'I TO'TI'I I'2 



6. 



#+£.#+£ a — b , a + b 

—7* — i — t. — ; — 7 — h 



5 
2a + b 



10 ' 
« + b 



x -y x-y . 



3 


6 




& ,c . 11.1 1^ 

^ </' a b x y } y 


w 1 

"7'T 


l 



7. - 



Section 140. Steps in adding or subtracting fractions. 

In the addition or subtraction of fractions it is most eco- 
nomical to follow a definite procedure. A study of the 
examples in the previous exercise suggests the following 
steps or procedure for combining fractions. 

First step : If the fractions have the same denominators, write 
the algebraic sum of the numerators over the common de- 
nominator. 

What examples in the previous exercise suggest this, 

rule ? 

Second step : If the fractions have unlike denominators, 
change the fractions into equivalent fractions which have the 



The Use of Fractions with Letters 



289 



same denominator, and write the algebraic sum of the numera- 
tors over the common denominator. 

What examples suggest this rule ? 

Third step : The resulting fraction should be reduced to 
lowest terms. 

EXERCISE 118 
PRACTICE IN ADDING AND SUBTRACTING FRACTIONS 

Do as many as you can orally : 
and 



ajj c , b x 2 ,5 
1. Add : - and -; 



2. 



d 

and 



c j d 1 j p 

, -and — ; - aiKK 

ay y a* a* r r 



_ a a-b_ x _ ± y_ ^ x _-y c±d^c_ 



d 



3. - and — ; - and 

b d y z 



4. a l±* and a 



a 
w. ab 

x 



a m 

1 be a j d 

— and — ; — and — 

ac 



m 



y be 



5. 



x+y 

: 2 + d 2 



and 



x + y 

e 2 - d 2 



6. 



7. 



10 and «-!±i- 2 



x+y x+y 

x+y x—y ,1/ 
and^ 



x—y x—y 3 3 

Add and check each of the following : 



3 



8 . *_±Z + *L 



v 



9. 



10. 



6 
a + b 



+ 



3 
a — b 



11. 



1x 



15 



12. « + c - + e - 
b d f 



b 2 a 2 + 2ab + b 2 



13. 



x — y 



y. 



14. Add 1 + ~ 

a b 



Think of the result which you 



obtain as a formula for adding any two such 
fractions. Write this formula as a word rule. 
Apply this formula to the addition of 



290 Fundamentals of High School Mathematics 



1 ,11,11 , 1 

-and-; -and-; -and-; 

1 A 1 1 ^ 1 1,1 

3 and 7 ; 8 and 5 ; i + >' 



15. Show that = 

a ab 



Think of this as a 



formula for finding the difference between two 
such fractions. Write the rule in words. 
Apply the formula to 

111111111111 



5' 5 4' 2 5' 



9' 3 7' x 



y 



16. Show that g + g = ^ + bx . Think of this as 

y by 

a formula for adding any two such fractions. 
State the rule in words. Apply this formula to 

3 5 2 3 12 8 5 

4 + 7' 5 + 4' 6*5' 3 + 2' 

2,3. a £_. £ a* 
9 4' b d y d 2 b 2 

17. Show that- = -*£— Think of this as 

by by 

a formula, and apply to : 
3272821323 b 2 x 2 





4 5' 3 6' 


7 


9' 


4 


5' 


9 4' <: y 


18. 


a 2 a 7 a 
5 I 15 








21. 


3 3 1 
a 3 2a a a 


19. 


2b b 5b 
7 3 21 








22. 


4 + A + l 

j/ 5_^ j/ 2 


20 


4 + J 5_J3 

^r ;tr 2 ;tr 3 








23. 


b c a 



The Use of Fractions with Letters 



291 



24. 
25. 
26. 
27. 
28. 
29. 
30. 

31. 

40. 
41. 
42. 

43. 



x+2 



+ 



5x 



x-2 {x + 2){x-2) 



5 6 



£ + 2 £-3 

3 + 5 



x + 4 .r + 3 
4 2 



;r+2 ^-+3 

4 + 3 



b-5 b + 2 
x+2 x-6 

5 3 

2^ + 1 3£-4 

4 7 

<z — £ /: — a 



c 
3 + f 
5 + f 
1 + f 

x 



45. fl + f 

46. I4-- 



47. 1 - 



44. 1 + 



3 7 



48. 1 



55. 



56. 



1 + 1 ; + " 

a c 
f F 



32. 
33. 
34. 
35. 
36. 
37. 
38. 
39. 



5 


3 




x-2 


,r+6 




8 


5 




y + 5 


y + 8 




a 4-2 


0-I 




* + 3 


« + 7 




,-l_ 


j-2 




j + 4 


7 + 5 




5 


, 6 




^• 2 -4 


1 x + 2 




X 


, 3 




z*-9 


'*-3 




4 


1 


X 


* 2 +5. 


r-f 6 ^ 


-4-2 


*+7 


.r + 5 



* 2 + 8.r4-15 4^+3 



49. 1 + 



b* 



50. 



c 

1 



51. ^r4- 



52. y-± 

53. c 



54. 1 + 1 



58. - 



59. 1 



a + b a 



-b 



a + b 

y 



x+y 



57. 



*4-2 * + 3 



292 Fundamentals of High School Mathematics 

60. If a certain pipe can fill a tank in 12 minutes, 
what part of the tank can it fill in 1 minute ? 
What part could it fill in 1 minute if it requires 
t minutes to fill the tank ? 

61. A tank can be filled by one pipe in / minutes, 
' and by another pipe in T minutes. What part 

of the tank can be filled in 1 minute if both 
pipes are open ? Express this as one fraction. 

62. If A = - and B = — , what does A + B equal? 

x Zx 

What does A — B equal ? 

63. Find A +B if A=—^— - and£ = — ^— • 

x + 5 x — i 



HOW TO MULTIPLY ALGEBRAIC FRACTIONS 

Section 141. Multiplication of fractions. You have 
seen that addition, and subtraction in algebra are per- 
formed exactly the same as in arithmetic. Multiplication 
of fractions, also, is performed in exactly the same manner. 

The following exercise will illustrate. 

ORAL EXERCISE 119 

1. 5xf; 6 x |; 7 x |; 5 xf; 8 x |; 12 xf 

2. *.f; * v f; 2* -J; 3j.f; 5*.f; } - Ay, 
3 - ixf; fxf; * xf; ixf; f x f 

a c x a a c 2b Sa 
6 X ~d' y*V' 7 1' 37 X 4^' 

10. 



5 4 y JUL . 6_ Y - 8 - • -9 * •£ ' 1 2 y 1Q. 
°' 5 X 2>4 X 12>8 X 3>^ X ^ 

_ ab c be ^d x w 
6. — x — ; — -x-; -• — • 
c a ad c y y 



The Use of Fractions with Letters 293 

Section 142. Steps in multiplication of fractions. In 

solving the examples in the previous exercise, you made 
use of the following steps or rules : 

To multiply a fraction by an integral expression, multiply the 
numerator by the integral expression and write the result over 
the denominator. For example : 

5 5 5 c c c 

To multiply a fraction by a fraction, multiply the numerators 
together for a new numerator, and the denominators together 
for a new denominator. For example : 

3 7 _ 3 • 7 _ 21 , a c_ a- c _ac 

5 8~5.8~40'6 d~b-d~bd 
To reduce the result to lowest terms, divide numerator and de- 
nominator by any factor that is common to both. For example : 
2 





*** = 

7 9 


5 


x^ 
9 


9 


a 


x» = 

C 


_a-V_a 


EXERCISE 120. 


MULTIPLICATION OF FRACTIONS 


1. 


X 
yX- 

y 




3. 


be > 


.b 
c 




5. 4tf X / 


2. 


c 
ax - 

a 




4. 


5x 


5 




6. bc 2 X- 
bc 


7. 


2 1 

xy 










12. 


a b 
b'c 


8. 


a + 


b 








13. 


xy a z b^ 
ab x 


9. 
10. 


1 

mn — 
in 

3-4 


, 








14. 
15. 


m^n y s 
xy 2 nfi 
2 ■ 3 2 35 
5.7 2 2 • 3 


11. 


aW • -V 

abc 










16. 


x±2 x+S 

X + Z x + 1 



294 Fundamentals of High School Mathematics 



17. (x + 2)(x-2) 



18. (y+S)(y-$). 



19. 

20. 
21. 
22. 
23. 



x+2 ~ /N " ' y + Z 

Check by letting x — 1 and y = 3. 



2^ 2 6 j/ 2 

Multiply 2 • 3 • 4 by 5. 
Multiply 5 2 - 3. 4 by 2. 
Multiply aHc by b. 
Multiply xy 2 w by ;r 2 . 

24. State the rule for multiplying a product by a 
number. 

25. Give a formula for the product of any two frac- 
tions ; of any three fractions. 

26. How do you affect the value of a fraction by 
multiplying its numerator by a number? by 
dividing its denominator by a given number ? 
by multiplying both its numerator and denomi- 
nator, by a given number? by dividing both 
numerator and denominator by a given number ? 

27. Is the value of a fraction changed by adding 
the same expression to its numerator and de- 
nominator ? Illustrate. Is its value changed 
by subtracting the same expression from its 
numerator and denominator ? Illustrate. 

28. Do you change the value of a fraction by squar- 
ing both numerator and denominator? Illus- 
trate. 

29. Tell what has been done to the first fraction in 
each of the following, to give the second frac- 
tion, and then tell whether or not the value has 
been changed : 



The Use of Fractions with Letters 295 



/ x a ac , x x x* / N a a 



y- 


■5 


y- 


4 


b + t 




b-c 




1 




x — 


y 


1 





b be y y b b — x 

V ' b b + 1 V ^p !£/ ^ ' ^ tf 

30. Summarize all the operations that may be per- 
formed upon a fraction without changing its 
value. 



31. iX| • {x + 8>(* - 3) 35. O + 6)0- - 4) • 

32. |±|.(£ + 2)0-1) 36. (d+c)(b-c)- 
— 1 

33. £=|.(,+ 2)(<» + 8) 37. (*+y)(x-y) 

c + 2 

34. ^=4 (^- 4 )(- r - 1 ) 38 - (^ 2 + 5^ + 6). 
.r — 4 x 4- 3 

39. Multiply tf 2 + 8tf + 6 by ^-±|. First, factor 

CL -\- L 

a 2 + $a + 6. 

40. r 2 + 6 y + 9 by 2±| 43. r 2 - 9 by ^±^ 

^ j + 3 ^ r+ 3 

41. *a - 3* - 10 by ^| 44. /> 2 - 6p by - 

^ x—5 P 

42. ^ 2 -f 6 <: + 5 by C —^- 45. £ 2 - 8 b by ^±-? 

y ^ + 5 J b 



HOW TO DIVIDE FRACTIONS 

Section 143. Division of fractions. In arithmetic, you 
learned how to divide one fraction by another, i.e. to in- 
vert the divisor, and proceed as in multiplication of frac- 
tions. 



296 Fundamentals of High School Mathematics 

For example : 

2_5 = 2 x 7 = 14. 3^ 5= 3 X 1 = 3^ 
3 7 3 5 15' 4 4 5 20' 

In the same way, in algebraic division, we invert the di- 
visor and multiply. 



EXERCISE 121 

Do as many orally as you can. 

-.tv^ * u b b , b 3 , 6 6 u 3 

1. Divide: -by-; g by -; - by -; -by-. 

«t^--j a,cx,5bc-,b 

2. Divide: - by -; - by -; — by -• 

b ay y ad a 



3. Divide : 



x -\-y 

2 



u x — y 

by -Y- 



a + b 



by 



by 



4. 



5. 



a + b 
ab m be 
xy xy 
c 2 d . <?d 2 
ab a 2 b 

Sx 2 y 



a — b 



7. - 



a _^_b 

1 '' c 
l^_x 

x y 



a A 



9. 



10. Divide 



2xy 



2 by I 



y = 2, a = 3, and £ = 4. 
10** 15 * 2 



11. 



12. 



13. 



14. 



14 j/ 2 ' 21 j/ 5 
12 be 2 18 W 



15 ^ 

2*j/ 



5 a 
4x 2 y . 
51 £ ' 17 ^ 
(2^) 2 . 18a 3 £ 

25 c 2 ' 100 ^ 



7 j 
Check by letting x = 1, 



15. 



16. 



17. 



18. 



(x+y)(* 


-JO.. 


x+y 


(a + b)(a 


-') 


a — b 


x 2 — y 2 


* — y 




a 2 -b 2 ' 


a + b 




5 ' 4 






(4^) 2 . 


Sa 2 




(2j)3 


6x 2 yZ 





The Use of Fractions with Letters 



297 



19. 



x 2 + 6x+8 m x +2 
a 2 + 4 a + 4 ' a + 2 



20. 



25 . x+5 
-3 



-9 



21. Tell by what the first expression in each of 
the following must be divided to give the second 
expression. 

(e) abxy, y 
(/) lwh t h 

(A) irR 2 , R 2 



(a) —%>x,x 
{b) bey, y 

W % y 



(i) c 2 x, x 
(j) NTx, x 
(k) IFt, t 
(/) m{a + b)y,y 



SUMMARY OF CHAPTER XIV 

The following points should be mastered by the study 
of this chapter : 

1. An algebraic fraction is a quotient, or an indi- 
cated division. Its terms are numbers, with 
letters. 

2. The terms of a fraction may be multiplied or di- 
vided by the same expression without changing 
the value of the fraction. 

3. Fractions are reduced to their lowest terms by 
dividing both terms by all factors which are 
common to both. Hence, to reduce algebraic frac- 
tions, it is usually necessary to factor both terms. 

4. Only like fractions can be added or subtracted. 
If we wish to add or subtract unlike fractions, 
we must change them into equivalent fractions 
which have the same denominator. 

5. The most convenient denominator to which frac- 
tions must be changed (in order to add or subtract 



298 Fundamentals of High School Mathematics 

them) is the expression of lowest degree which 
will contain each of the separate denominators. 
6. Multiplication and division of algebraic fractions 
are performed exactly as in arithmetic. 



REVIEW EXERCISE 122 



Think of 



1,1 a\-b ,1 1 
+ - = — —~ and 



a b ab a b ab 

as formulas for the addition and subtraction of a 
special kind of fractions. Apply these formulas to 
the following : 



(«H + i (/)*-tV 



(/) 



b 2 ~ 


~c* 


1 

ab 


1 

cd 


1 


, 1 



«1 + 1 ie)l-\ 

2. Think of 

# , £ _ fl^H- <fc a a _ c — a d — be 
b d bd b d ad 

as formulas for the addition and subtraction of frac- 
tions. Apply these formulas to the following: 

Wi+f (/)A-t U)Z-- 

W X 

lw_ _3 
2 x 

3. Change to a single fraction : — + -- + -. 



Wl + f (*)*-* 

Wi + i Wf-! W 



The Use of Fractions with Letters 299 

4. Reduce to lowest terms : ; ; 

2 d • o J • 5 a b b 6 c 

2 2 x5 3 
4 x 25 x 23' 

5. A = and B = . Which expression, A or 

x y x — y 

B, has the greater numerical value, if x = 5 and 

7 = 3? 

6. Does ^ + f 2 = a + £ ? Check. 

a + b 

7. If y4 = I and i? = |, what is the ratio of A to B ? 

8. Why should the chapter on Products and Factors 
come before the chapter on Fractions ? 

9. Solve with two unknowns ; two thirds of a certain 
number, increased by f of another number, gives 
25; the sum of the two numbers is 35. Find each 
number. 

10. The value of a certain fraction is f ; the sum of the 
numerator and denominator is 25. Find the fraction. 
\2x — y = 8 { x + y = b — a 

I ;r— 2=j/-fl \2x — y = — 5a — b 

13. The sum of two numbers, x and y y is a> and their 
difference is b. Find each number. 



CHAPTER XV 

LITERAL AND FRACTIONAL EQUATIONS 

Section 144. A new kind of equation : literal equations. 

Thus far in the equations which you have solved, the only 
letter used was the letter used to represent the unknown. 
All other numbers were arithmetical or ordinary numbers. 
Now, we shall study equations which, in some cases, will 
not contain any ordinary numbers. That is, the known 
numbers may be represented by letters instead of being ex- 
pressed in ordinary numbers. For example, consider the 
equation, or formula, for the volume of a rectangular box, 

Iwh = V. 
There are cases in which we wish to restate this equation 
in a new form so as to give the value of /, that is, the 
length. In such a case / would be the unknown number. 
In other cases we might wish to restate the formula 
so as to find the value of w, that is, the width ; then w 
would be regarded as the unknown number. If we con- 
sider / as the unknown in the equation, then w, h, and 
Fare regarded as known numbers and are treated just as 
if they were arithmetical numbers. 

Two illustrations will make this clear. 

Illustrative example. 

Solve for / the equation, Iwh = V. 
Solution : luth = V. 

Dividing each side of the equation by wh, gives 

wh 

By what would'you have divided to solve for w ? for h ? 

for V? 

Second illustrative example. 

• Solve for x the equation, 

ax + bx = c. 



Literal and Fractional Equations 



301 



(1) Factoring the left side, to get x by itself gives 

x(a + b) = c 

(2) Dividing each side by a + b gives 

* = -?-. 

a-\- b 

The next exercise will give practice in solving easy 
equations with letters. 

EXERCISE 123 — ORAL WORK 

Solve each of these equations : 

Consider the letters toward the end of the alphabet, either 
x, y, or z for example, as the unknowns, and those toward 
the beginning as the knowns, unless otherwise directed. 



1. 


y-b = § 


12. 


by — be 




2. 


x-b=2b 


13. 


ex — c 2 d 




3. 


w -\-^a — l a 


14. 


4 ax = 12 ac 




4. 


x — 2>c = — 5 c 


15. 


3by = 9bc+6b 




5. 


y -f- 5 a = — 4 # 


16. 


3x-6a=9b 




6. 


x — b = c 


17. 


2y-±b = 2c 




7. 


y=2a=b 


18. 


ax = a 2 + 2 ab 




8. 


w-3c = 2b 


19. 


5 ay + 4 ac = 14 ac 




9. 


2x = 4:b-6c 


20. 


bx—bc — ab 




10. 


5x-Sc = llc 


21. 


x{a + b)— a + b 




11. 


ay = a 


22. 


y{c — d) — {c — d)(c - 


-d) 



23. w(a + b) = (a + b)(a - b) 

24. x{a + b— c)= 4 a(a -\- b — c) 

25. y(a + £)(> - b)= b{a - b)(a + b) 

EXERCISE 124 

Solve for the unknown in each of the following, and check : 



302 Fundamentals of High School Mathematics 

Illustrative example. 

Solve 4y— b = y + 5b. 

Transposing, 4z/ — y — bb + b. 

Collecting, 3y = 6b. 

Dividing by 3, y = 2 b. 

Checking, 4-2b-b = 2b-5b. 

8b-b = 2b + 5b. 
7b = 1b. 

1. Zx — e = x + 9e li. ^—2^ = 5^ — 3^ 

2. 4j/ + £=.y + 4£ 12. j/ + 2(>- b)=±a- b 

3. x — a = a + b 13. 2(j/ — b)= 6 be 

4. 4c by + 2 b = 14: b 14. — (^ — c)= — 3 c 

5. <2;tr -|- #£ = <2# 15. — b(x — 3)= lb 

6. or — c 2 = be 16. 2 bx — 3 ab = 6 ab — bx 

7. 6^ + £ 2 = 7£ 2 17. \bx = ±b-\bx 

8. — rt^r -+- ab = — #£ 18. x(a + b)—2a = 2b 

9. be— by = ab 19. j(V — <z) = %c — <z) 
10. car — be = c 2 20. 2 dry 4- ac = ab 

21. Illustrative example, ax — ac = be — bx 

(1) Transposing, ax + bx = be 4- ac. 

(2) Factoring the left side, to get x by itself gives 

x{a + b)= ac + be 

(3) Dividing by a + b, gives 

x — gc + bc — c ( g + &) — c 
a + & ~ a + & 

Checking : a • c + b . c = ac + be. 

22. ay—by = ae—bc 27. 2ay—by = {2a—b)(a — b) 

23. 3 cr + 2 for = 6 £ + 4 £ 28. 4 a* — ;tr = 16 a 2 — 1 

24. ax — bx = ae — be 29. by — 2y = b 2 — 5 b + 6 

25. a— / = 4# + 4 30. ^^4-5^- =# 2 + 8^ + 15 

26. by+y=3b + 3 31. &tr + 4 = * -}- 4 £. 



Literal and Fractional Equations 



303 



Section 145. Equations containing fractions with literal 
denominators. We have already solved many fractional 
equations in which the denominators were ordinary arith- 
metical numbers. (For example, in Chapter II, such ex- 
amples as f/ + fi> =/ + 5.) Now we shall take up the 
solution of equations which contain fractions with literal 
denominators. 

Recall how you solved equations such as 

Note that you found the most convenient multiplier, 10, 
and then got rid of all fractions in the equation, by multi- 
plying each term by 10. 

In the same way, you solve such equations as 

x . x , 7 

- + - = a + b. 
a b 

That is, you find the most convenient multiplier to use in 
order to get rid of fractions. Why is it ab in this equa- 
tion ? Then the equation is of the same form as those in 
the previous section. 
Illustrative example. 



Solve 



* + I = a + b. 
a b 



(1) Multiplying each term of the equation by ab gives 

bx + ax = a 2 b + ab 2 . 

(2) Factoring the left member to get x by itself gives 

x(b + a) = a 2 b + ab 2 . 
(3) Dividing by b + a gives 

x = a 2 b + ab 2 = ab(a + b) = ah 
b + a b + a 

Check : Let the student check the example. 



304 Fundamentals of High School Mathematics 



EXERCISE 125 

Do Examples 1-15 orally. 



1. 


4 = 2, 

X 


6. 


— =2 


11. 




2. 


§=1 

y 


7. 


^ = 4 
Sb 


12. 


2£ 


3. 


l£-6 

2x 


8. 


c - = b 

X 


13. 


a 2 


4. 


x - = b 
a 


9. 


b 
- = a 

y 


14. 


2 £ 


5. 


-f =3 

2b 


10. 


a _ 

X 


15. 


^ + 1_5 
a a a 



Each of the equations in Examples 16 to 27 is a practical 
formula. These formulas are used commonly in various 
kinds of work. It is often necessary to be able to solve 
them for any letter. The rest of this exercise, therefore, 
will give you practice in solving such practical formulas. 

16. Solve A = bh for b ; for //. 

17. Solve V — Iwh for /; for w. 

F 

18. Solve C=-^- for F; for R. 

R 

19. Solve d—rt for r\ for t. 

20. Solve i =prt for/ ; for t. 
D 2 N 



21. Solve P 



2-5 



for iV. 



22. If - = e, what does b equal ? 

b 

23. If a{x + c)= be, what does x equal ? 

24. Solve A = ^ f or ; for A. 



Literal and Fractional Equations 



30S 



25. If A= *( a + d \ what does h equal? What 

does a equal ? 

26. Given the formula a =p(l + rt\ solve for /, 
and then for r. 



TIMED PRACTICE EXERCISE H 

Practice under time to see how many examples you cam 
do in 5 minutes. The very rapid pupils in 100 cities did 
about 20 ; the average pupils did about 12. 



l. i = pn Solve for r io O 

forZ 

13. 



2. s = — Solve 

EI 

— : Solve for ii 

7^ 



A 



Solve for z; 



3 - *=¥ 

M 
4. (7= 



14. 



— — - Solve for K 15. 

5 M= ™ Solve for £ 16. 

o 

6. r = ^ Solve for / 17. 

a 

7. F= Z 0% Solve for W 

s ia 

8. z> = — Solve for T 

T 19. 

9. j = — Solve for a 20. 

r 

10. R = EL-— I Solve for W 21. 

11. z; = — Solve for b 22. 

o 



P = ahw Solve for ^ 

E 
c = -= Solve for i? 
E 

PL 

E — —— - Solve for P 

L = m ~<? Solve fovM 
t 

bd* 

3 



/ 



Solve for b 



E 2 



JWfa 



Solve f or h 



v = irLr 2 Solve for L 

A = ~ Solve for M 

M 

M=^I Solve for 5 1 



/ 



_gm 



Solve ioxg 



306 Fundamentals of High School Mathematics 

Section 146. Equations containing fractions with bi- 
nomial denominators. In the previous fractional equations, 
no denominator contained more than a single term. There 
will be occasions in your later work when you will need 
to be able to solve fractional equations in which the 
denominators contain expressions with two letters, i.e. 
binomial denominators. For example, suppose you were 
to solve the formula 

C=-^- for R; 
R + r 

it would be necessary to get rid of a binomial denominator, 
R + r. Therefore, we shall next learn how to solve such 
equations. Let us illustrate with the equation 

x = 1 

x+5 3' 

No new principle is required to get rid of fractions in this 
equation. The only difficulty is that involved in select- 
ing a convenient multiplier, i.e. in finding the lowest com- 
mon multiple of the denominators. In this example, we 
must get rid of the denominator x + 5, and of the de- 
nominator 3. Therefore we must multiply each term of 
the equation by 3(x + 5). Why ? 
Illustrative example. 

Solve — *- =1. 
x + 5 3 

(1) Multiplying by 3(x + 5) gives 

(2) Reducing gives 

3x = x + 5 
or 

x = 2.5. 

Check : = - ; or — = - or - = -• 

2.5 + 5 3' 7.5 3 3 3 



Literal and Fractional Equations 307 

EXERCISE 126 

PRACTICE IN SOLVING EQUATIONS WHICH CONTAIN FRACTIONS WITH 
BINOMIAL DENOMINATORS 



1. 



8. 



x + 5 


2 


y _ 


3 


7 + 2 


2 


x-2 


3 


x + S 


8 


b-1 


1 


b + 2 


4 


5 


2 


x+S 


;r 


10 


4 


j + 7 


JP 



X 


6 


x-S 


3 


2 
j+ 4 


_5 


_3 
y + 


1 
: 1 


,r-l 
*-2 


■H 


4 _ 1 
;r ;tr — 


3 


6 
5;r # 


-2 
-3 



3. = - 9. 



10. 



5. -=■-=■ 11. 



6. - = - 12. 



Second illustrative example. 
Solve *z^ = *— *1 

x + 3 x + 5 

(1) Multiplying each side by (x + 3)(x + 5) gives 

(2) Reducing gives (jc + 5)(x-4) = (x + 3)(x— 16). 

(3) Multiplying, x 2 + x — 20 = x 2 — 13 x — 48. 

(4) Transposing, x 2 + x — x 2 + 13 x = 20 — 48. 

(5) Collecting, 14 x = - 28, 
or x = - 2. 

Check: -2-4 = -2-16 r -6^-18 or _ 6 ^_ 6> 



2 + 3 -2 + 5 1 3 

!±» 14. iZ 



13. £±A = Z±1 14 J/-l_ J/ + 3 



308 Fundamentals of High School Mathematics 



15. 



16. 



3 b-2 



£ + '5. b + 2 

x + 1 _ x — 4 
x + 4 ~~ x + 2 



17. 



4 4 
18. ^-f-i— = 

;r ;r + 2 



2fr-3) 3(4-^) = 59 
5 2 5 



20. 



21. 



1 + x 



4a 



;r + a x—a {x — a){x — a) 
2 4 3 



a.' 



22. 1 + ^*- 



x — 3 ;r 2 
- = 1. 



9 



23. Solve (7 



for R (a formula used in elec- 



R + r 
tricity). 

24. Solve for / and n : s — — — -J — 2.". 

2 

25. Solve C = $(F— 32) for F(a. formula used in chang- 
ing thermometer readings from one scale (Fahren- 
heit) to another scale (Centigrade)). 

26. Solve the formula = — for/; for/. (This 

q p f 

is a formula used in studying lenses.) 



27. If A 



x + 5 



-and B = 



1 



-5 



, find A + ^ and ^4 - B. 



28. Solve 7? 



C_ 

N 



+ 1 for N. 



29. Solve f or A : 1=1 + 1. 



Literal and Fractional Equations 



309 



STANDARDIZED PRACTICE EXERCISE I (TIMED) 

Practice on this exercise until you can reach the 
standard, 9 examples right in 10 minutes. 



1. 


2 y - 5 y - 2 2 y - 3 \ 
3 5 15 


2. 


3 4 _ 


x - 8 .r - 10 


3. 


j + 3 j + 5 
j--8 - j--2 


4. 


£ + 4 £ + 1 b + 6 


£_7 b + b P -2d -35 


5. 


3j-2 „ 2j/-l l-3j 
4 3 12 


6. 


5 - 3 =0... 

x — 2 x — 6 


7. 


/ + 2 _ /> + 3 
t—1 t—4' 


8. 


c + q ' c + 2 1 c + 3 
<:_8 c +6~ c 2 -2c-4;8 ' 


9. 


2w- 1 1 Sw - 2 _ w + 3 
3 7 21 


10. 


4 2 -0.... 




.r - 3 x - 9 


11. 


r + 5 _ r + 4 
r — 6 r — 1 


12. 


y + 2 j/-f 4 _ 9j/+7 


7-7 _y + 10 _ / + 3j/ - 70. ' 



310 Fundamentals of High School Mathematics 

THE CONSTRUCTION OF GENERAL FORMULAS 

Section 147. Now that we have learned how to solve 
literal equations we can construct formulas for a great 
many new types of problems. Let us illustrate and solve 
a few of these types. 

Illustrative example, a. particular problem; that 

is, problem with arithmetical numbers . The sum of two numbers 
is 20, and the larger is 9 times the smaller. Find the numbers. 

Let x = smaller no. 

Then 9 x — larger no. 

X + 9 x = 20. 
Solving, x = 2 and 9x = 18. 

B. GENERAL PROBLEM ; that is, problem in which the num- 
bers are all literal. The sum of two numbers is S, and the 
larger is m times the smaller. Find the numbers. 



Let 


x = smaller no. 


Then 


mx = larger no. 


Solving, 


mx + x = s, 




x(m + l)=s, 


or 


x- s • 


m + 1 


Therefore, 


mx — • 



m + 1 

Note that the particular problem is a special case of the 
general problem ; i.e. when s = 20 and m = 9. Thus, the 

expressions and may be regarded as general 

m-\-l m + 1 

expressions or formulas, for finding two numbers when 
their sum and ratio are known. In the following exercise, 
you will first solve a particular problem, with arithmetical 
numbers, and then solve a general problem which repre- 
sents all the special problems of that type. 

What are the numbers if s — 30 and m — 4 ? If s = 100 
and m = 9 ? 



Literal and Fractional Equations 311 

EXERCISE 127 
PRACTICE IN CONSTRUCTING GENERAL FORMULAS 

1. (a) Particular problem : The sum of two num- 

bers is 16 ; the larger is 4 more than the 
smaller. Find each number. 
(b) General problem : The sum of two numbers 
is S ; the larger is a more than the smaller. 
Solve for the two numbers, i.e. make a for- 
mula for finding each number in this type of 
problem. Evaluate when >S = 40 and a is 10. 

2. (a) The sum of two numbers is 32, and their dif- 

ference is 8. Find each number. 

(b) The sum of two numbers is S, and their dif- 
ference is d. Find each number. 

(c) Read the formulas you obtain in (b) as rules 
for finding the numbers. 

(d) Evaluate when S = 52 and d = 10. 

3. (a) The sum of two numbers is 9 ; 10 times the 

smaller equals 5 times the larger. Find the 
numbers. 
(J?) The sum of two numbers is vS, and m times 
the smaller equals n times the larger. Find 
each number. 

(c) Read the formulas in (b) as rules for finding 
the numbers. 

(d) Evaluate when 5 = 12, m = 6, and n = 2. 

4. (a) A rectangle is 10 ft. longer than it is wide ; its 

perimeter is 72 ft. Find its length and width. 
(b) A rectangle is b feet longer than it is wide ; 
its perimeter is / feet. Find its length and 
width. 



312 Fundamentals of High School Mathematics 



(c) Read the results in (b) as rules for solving 
any problem of this type. 

(d) Make up a particular problem and solve it 
by the use of these formulas. 

5. (a) The sum of two numbers is 14 ; three times 

their sum is equal to 21 times their difference. 
Find each number. 

(b) The sum of two numbers is 5 ; m times their 
sum is equal to n times their difference. 
Find each number. 

(c) Read the formulas in (b) as rules for solving 
problems of this type. 

(d) Make up a particular problem which you 
can solve by these formulas. 

(e) In problems of this type, can m — n} Can m 
be greater than n ? Can it be smaller than n ? 

6. {a) Separate 20 into two parts, such that the 

quotient of the larger by the smaller shall be 
f . Use x for the smaller. 
(b) Separate n into two parts, such that the 
quotient of the larger by the smaller shall be 



(c) Read the formulas which you obtained in (b) 

as rules for solving problems of this type. 
(d) Make a particular problem which belongs to 
type. 

(e) Can a = b? Can a be smaller than b ? Why ? 

(a) Think of some number ; multiply it by 3 ; 
add 9 to the result; multiply the last result 
by 2; divide the last result by 6; subtract 



Literal and Fractional Equations 313 

the number thought of in the beginning ; the 
result must be 3. Can you tell why ? 
(b) Think of some number, as n ; multiply it by 
a; then add 3a; then multiply by 2; then 
divide by 2 a ; then subtract the number 
thought of in the beginning. Show that the 
result must be a. Can you make up other 
problems similar to this ? 

EXERCISE 128 
WORK AND RATE PROBLEMS 

1. Illustrative problem. One contractor, Mr. A., has the 

facilities for building a house in 15 days ; another con- 
tractor, Mr. B., has the facilities for building it in 20 days. 
The owner was very anxious to have the house completed 
at once ; so he had both contractors work. How long 
would it take them, working together, to build the house ? 

A can do — of the work in 1 day. 
15 3 

B can do — of the work in 1 day. 
20 

Both together can do — V — of the work in 1 day. 
15 20 

Let x = no. of days required for both to do the work. 

Then - = what both can do in 1 day. 
x 

This gives the equation 1 = -• 

6 H 15 20 x 

Multiplying by 60 x, 4 x + 3 x = 60, 

or x = 8f days for both to build house. 

2. Contractor A can do a certain piece of work in 
10 days ; contractor B can do the same job in 
8 days. How long would it take both working 
together to do the work ? 

3. The morning issue of a paper can be printed on 
one press in 5 hours, and on another press in 4 



314 Fundamentals of High School Mathematics 



4. 



5. 



9. 



hours. How long would be required if both 
presses were used ? 

A painter can paint a house in 12 days ; his ap- 
prentice would require 20 days. How long 
would both together require ? 
A tank can be filled by either of three pipes. 
Pipe A can fill it in 20 minutes ; pipe B can fill 
it in 25 minutes, and pipe C can fill it in 50 min- 
utes. How much time would be required if all 
three pipes were opened at the same time ? 



6. Solve for x : 



1 1 A* 

- = -,and- 

5 x 4 



= 1. 



4 5 x 4 5 

A can do a job in a days which B can do in b 
days. Both working together can do the work 
in how many days ? 

A works twice as fast as B. How long will it 
take both together to do what B alone can do in 
10 days? 

A can do a piece of work in a days, B in b days, 
and C in c days. In how many days can all 
working together do it ? 



PROBLEMS BASED UPON FRACTIONS 

EXERCISE 129 

Illustrative problem. The numerator of a certain frac- 
tion is 5 smaller than its denominator; if 1 be added to the 

numerator, the value of the resulting fraction becomes -• 

Find the fraction. 

Let d = the denominator. 

Then d — 5 = the numerator, 

d-b 



and 



= the fraction. 



Literal and Fractional Equations 



3i5 



By the condition of the problem 




d - 5 + 1 _ 1 




d 2 


t\r 


d-4 1 


UI 


d 2 


Solving, 


2d-S = d. 




d = 8, the denominator. 




d — 5 = 3, the numerator. 




f = fraction. 


Check : 


3+A = i ; or4 = l. 
8 2' 8 2 



2. The numerator of a certain fraction is 4 smaller 
than the denominator ; if 1 be added to the nu- 
merator, the value of the resulting fraction be- 
comes |. Find the fraction. 

3. The denominator of a fraction exceeds its numer- 
ator by 3 ; if the numerator is decreased by 3, 
the value of the resulting fraction becomes J. 
Find the fraction. 

4. The numerator of a certain fraction exceeds its 
denominator by 2. If the numerator be de- 
creased by 1 and the denominator increased by 
1, the value of the resulting fraction becomes 1. 
Find the fraction. 

5. The denominator of a certain fraction is twice as 
large as the numerator. Find the value of the 
fraction. What is its value if the denominator 
is three times as large as the numerator ? 

6. The value of a certain fraction is | ; if the 
numerator and denominator each be increased 
by 2, the value of the resulting fraction will 
be i. Find the fraction. (Use two unknowns, 
n and d.) 



316 Fundamentals of High School Mathematics 



8. 



The value of a certain fraction is -| ; if the nu- 
merator be increased by 1 and the denominator 
decreased by 4, the value of the resulting fraction 
will be §. Find the fraction. 
Make up a problem similar to one in this list. 
Your teacher will have the class solve it. 



SUMMARY OF CHAPTER XV 

1. A literal equation is one in which some or all of 
the numbers are expressed by letters. 

2. By transposing, all terms which contain the un- 
known letter are placed on one side of the equa- 
tion ; all other terms on the other side. 

3. If more than one term contains the unknown, we 
must then separate the unknown factors from 
the knowns, by factoring. This gives the coeffi- 
cient of the unknown, by which each side of the 
equation must be divided. 

4. Literal equations may contain binomial denomina- 
tors. If so, we get rid of fractions by multiplying 
each term in the equation by an expression which is 
the least multiple common to all the denominators. 



REVIEW EXERCISE 131 

1. B weighs § as much as A. The distance be- 
tween them on a teeter board is 24 feet. How 
far is A from the fulcrum if they balance each 
other ? 

2. A can do a piece of work in 4 days, B in 6J 
days. How many will they require, working 
together ? 



Literal and Fractional Equations 317 

3. An auto tourist made a trip of 120 miles, at a 
certain rate ; on the return trip he increased 
his rate 5 miles per hour, and required 4 hours 
less time. Find his rate going. 

4. What number must be subtracted from the de- 
nominator of the fraction J-J to make the value 
of the fraction -^ larger ? 

5. An athlete can run two and one half times as 
fast as he can walk. Find his speed in yards 
per second if he can run 100 yards in 12 less 
seconds than he can walk it. 

6. A football player started toward his goal with 
the ball, for a touchdown. An opposing player 
was 10 yards behind him, but could run 2 yards 
a second faster. The first player had gone 30 
yards when he was overtaken by the second. 
Find the rate of each, assuming that they 
started at the same time. 

7. The sum of two angles is 140°, two thirds of the 
smaller equals one half of the larger. Find each. 

8. A common number trick is : Think of a number ; 
add 7 ; double the result ; subtract 8 ; tell me 
your answer, and I will tell you the number you 
thought of. Can you explain why this is so ? 

9. A train running at the rate of r miles per hour 
required t hours to make a trip with u stops of 
s minutes each. What distance did it travel ? 

10. (^ + l)(^ + 2)-(^r-3)(^-4)=0. Find x 
and check. 

11. A man receives w dollars each week in the 
year ; his expenses average / dollars per week. 



318 Fundamentals of High School Mathematics 

He takes a two weeks' vacation, during which 
his expenses are increased 25 dollars per week. 
How much will he save in a year? 

12. Factor: (a) 30 * 2 + 61 x + 30. 

(V) 6* 2 - 13* + 6. 

13. A boy left a certain place, riding his motorcycle 
at the rate of 12 miles per hour; two hours later 
a second boy started to overtake him, riding at 
the rate of 16 miles per hour. In how many 
hours will he overtake him ? Solve graphically. 

14. If A should give B $ 6, they would have equal 
amounts, but if B should give A $1, A would 
have three times as much as B. How much 
has each ? 

15. Two unknown weights balance when placed on 
a teeter board, 8 and 10 feet from the fulcrum. 
If their positions are reversed, 54 pounds must 
be added to the lesser weight to make a balance. 
What are the weights ? 

16. A grocer has two kinds of sugar, one worth 
10 a pound, the other worth 12 ^. How many 
pounds of each must he use in a mixture of 50 
pounds worth $ 5.60 ? 

17. The admission price to a moving picture show 
was 10 ft for adults and 5 <f, for children. One 
evening there were 420 admissions and re- 
ceipts of 136. How many adults and how 
many children attended the show ? 

18. Solve both algebraically and graphically : 

}3*-7 = 10, 
6*=2j/ + 4. 



CHAPTER XVI 

HOW TO SHOW THE WAY IN WHICH ONE VARYING 
QUANTITY DEPENDS UPON ANOTHER 

Section 148. Quantities that change together. We have 
already seen that there are many illustrations of quantities 
that change together. The amount of money paid out for 
rent at $30 per month changes with, or depends upon, the 
number of months ; the time required to walk a certain 
distance, say 10 miles, changes as, or depends upon, the 
number of miles one walks per hour. In other words, 
there are varying quantities which are so related that a 
change in the value of one of them causes a change in the 
value of the other. 

We could not continue our study of these varying quan- 
tities in Chapter VIII because we had not then learned how 
to deal with fractions and fractional equations. 

This chapter will deal with quantities which change 
together. In addition to what you already know about 
these varying quantities, we shall now study just how these 
quantities vary. For example, does an increase in the 
value of one varying quantity cause a corresponding in- 
crease in the related quantity ? Or does an increase in one 
varying quantity cause a corresponding decrease in the 
other ? Can these be expressed (1) graphically, or (2) by 
tables, or (3) by formulas ? These are the points which 
will be studied in the chapter. 

Section 149. Variables and constants. In our study of 
time, rate, and distance problems we saw that the distance 
traveled by a train running at any given rate changes or 
varies as the time which it has been running changes or 
varies. If a train runs at the rate of 40 miles per hour, its 
movement is described by the equation 

</=40/. 
3i9 



3 2d Fundamentals of High School Mathematics 

In this equation, d and t change as the train progresses 
along its journey. The value of d depends upon the value 
of t. This means that the distance and time are variables, 
while the rate is constant. 

Table 20 shows the tabular method of representing the 
relation between these related variables. This shows that 



Table 21 



If the no. of hrs. is 


1 


2 


3 


4- 


5 


8 


10 


15 


20 


then the distance is 


40 


80 120 160 200320 400 600 800 



a change in the time causes a change in the distance, or 
that a change in one variable causes a change in the related 
variable. 

EXERCISE 131 

1. In the above table, does an increase in number of 
hours always cause an increase in distance ? 

2. In the same table, find the ratio of each distance 
to its corresponding time. How do these ratios 
compare ? Do the ratios change ? 

3. A man buys a railroad ticket at 3 cents per 
mile. Show by the tabular method the relation 
between cost and number of miles traveled. 
Show from the table that as the distance in- 
creases the cost increases, but that the ratio of 
the cost to the distance does not change. What 
equation will show the same thing the table shows? 

4. Write the equation for the cost of any number 
of pounds of sugar at 9 cents per pound. What 
are the variables in your equation ? Tabulate 
the cost for 1, 2, 5, 8, and 10 pounds. Show 



How One Varying Quantity Depends on Another 321 

from the table that the ratio of cost to number 
of pounds does not change ; that is, it is constant. 

5. A rectangle has a fixed base, 5 inches. Its alti- 
tude is subject to change. Tabulate its area if 
its altitude is 4, 6, 8, 10, and 12 inches. Com- 
pare the ratio of any two values of the area with 
the ratio of the two corresponding values of the 
altitude. If one altitude is three times another 

altitude, the one area is 1 times the other 

area. Write the equation for its area. 

6. A bicyclist rides 10 miles per hour. Show, by 
three methods, the relation between the number 
of miles he travels and the number of hours 

required. In 6 hours he travels ? times as 

far as he travels in 8 hours. 

I. DIRECT VARIATION, OR DIRECT PROPORTIONALITY: 
THE STUDY OF VARIABLES WHICH ARE DIRECTLY 
RELATED 

Section 150. The problems in the previous exercise illus- 
trate direct variation, or direct proportionality. In each of 
the examples, one of the variables depended upon another 
variable for its value, and the ratio of any two values of 
one variable was equal to the ratio of the two corresponding 
values of the other variables. When two variables are 
related in this way, one is said to vary as, or to be directly 
proportional to, the other. Thus, to prove that two varia- 
bles are directly proportional, or vary directly, we must 
show that 

The ratio of any two particular values of one variable is 
equal to the ratio of the two corresponding values of the 
other variable. 



322 Fundamentals of High School Mathematics 



EXERCISE 132 

I. Illustrative example. A man earns $ 6 per day. Show 
that the amount he earns is directly proportional to the num- 
ber of days he works. 

Solution : 

(1) A = 6d. (We write the equation first, from the condi- 
tions of the problem.) 



(2) Tabulating : 



Table 22 



If dis 


l 


2 


5 


8 


10 


32 


then A is 


6 


32 


30 


48 


60 


72 



(3) Now select any two values of A, say 12 and 60, and the 
two corresponding values of d, which are 2 and 10. If the ratio 
of these two values of A is equal to the ratio of these two 
values of <f, then in the equation A = 6dwe know that A 
is directly proportional to d, or that A varies directly as d. 
Doesi| = T 2 o? Yes. 

Thus, A is directly proportional to d, or the amount a man earns 
at $ 6 per day is directly proportional to the number of days 

he works. This is often written -± = -± ■ A x means some 

A 2 d 2 
particular value of A, and A% means some other particular 
value of A; di and d 2 mean those particular values of d 
which correspond to the selected values of A 1 and A 2 . 

2. Write the equation for the area of a rectangle 
whose base is 10 inches. Then show by select- 
ing particular values of A and h that the area 
is directly proportional to the altitude. In other 
words show that 



How One Varying Quantity Depends on Another 323 

3. Write the equation for the circumference, C, of 
a circle whose diameter is D. Is C directly pro- 
portional to D ? Why ? 

4. Show that the perimeter of a square is directly 
proportional to the length of its side. 

5. Show that the interest on 81000 at 6% is di- 
rectly proportional to the time. 

6. x varies directly as y y and when ;r=10,j/ = 2. 
Find the value of x when y = 7. 

7. C varies directly as d y and when d— 12, c = 38. 
What is d when c = 72 ? 

8. Is your grade in mathematics directly propor- 
tional to the amount of time you spend in pre- 
paring your lessons ? 

9. Is the cost of a pair of shoes directly propor- 
tional to the size ? 

10. P and s represent the perimeter and side, re- 
spectively, of a square. What is the meaning 
of the statement 

•* 2 S 2 

Write this as a general truth. See Example 4, 
above. 

11. In Example T it was stated that the circum- 
ference, C, varies directly as the diameter, d. 
Would it be just as well to state it 

It is very often expressed in this form. Find 
C 2 if C x = 25, d x = 8, and d 2 = 12. 



324 Fundamentals of High School Mathematics 



12. 



13. 
14. 



15. 



C represents the cost of a railroad ticket, and 
d represents the number of miles traveled. 
What is the meaning of 



C 2 «2 

Express algebraically that x varies directly as y. 
If i represents the interest on a certain sum of 
money, and r stands for the rate, what is the 
meaning of the expression 

h-Zi} 

In a science class the topic of the evaporation 
of moisture was being discussed. The amount 
of evaporation that would take place in a given 
amount of time (the temperature remaining 
constant) was the particular problem. It was 
finally stated as follows : 
A 1 _ s t lm 

/in In 



What did they mean by this ? 

16. In studying the relation between the weight of 
water in a tank and its volume, a class used 
the expression 

Interpret this statement. Find W 2 if W 1 = 625, 
V 1 = 10, and V 2 = 15. 

17. The relation between the weight of a piece of 
wire and its length is expressed by the expression 



El 
W 



Ja. 



State this relation in words. 



How One Varying Quantity Depends on Another 325 



18. 



19. 



20. 



21. 



Express algebraically that the stretch, S, on a 
pair of spring scales, varies directly as the 
weight W. If a weight of 12 pounds produces 
a stretch of \ inch, "how much weight produces 
a stretch of -| inch ? 

A train runs at a constant rate of speed. State 
algebraically that the distance, d, which it 
travels, is directly proportional to the time, /, 
which it has been running. 
The number of gallons of gasoline consumed by 
an automobile is directly proportional to the dis- 
tance traveled. State the same fact symbolically. 
W varies directly as V, and when W= 12, 
V= 5. Find J^when ^=40. 



Section 151. One variable may vary as the square of, or 
the cube of, another variable. It frequently happens that 
two variables are so related that a change in one of them 
is accompanied by, or causes a greater change in, the 
other. That is, if one of the variables is doubled, the other 
related variable may be made four times as large, or eight 
times as large. An illustration will make this clear. 
Think of squares whose sides are of different length. 
Compare their sides, and then compare their correspond- 
ing areas. A tabulation will help. 

Table 23 



If s is 


2 


3 


4 


5 


6 


7 


8 


lO 


12 


15 


20 


etc. 


then A is 


4 


9 


16 


25 


36 


49 


64 


100 


144 


225 


400 


etc. 



Select any two particular values of A, such as 9 and 
!5, and the corresponding values of s, 3 and 5. Does 



326 Fundamentals of High School Mathematics 

the ratio of the two values of A, 9 and 25, equal the 
ratio of the corresponding values of s, 3 and 5 ? That 
is, does 

2 9 s=f? NO. 

That is, does —1 == -1 ? 

9 3 2 

Evidently not. But — does equal — ; that is 
£0 o 



A- 

A, 



does equal - 1 -* 

J 2 

This shows that the ratio of any two particular values 
of A is equal to the ratio of the squares of the correspond- 
ing values of s ; or, that the area of a square is directly 
proportional to the square of its side. Note the algebraic 
statement of this relation, 

A study of this illustration leads to the general state- 
ment : 

One variable quantity is directly proportional to, or 
varies directly as, the square of another variable 
quantity, when the ratio of any two particular values 
of the first variable is equal to the ratio of the 
squares of the corresponding values of the second 
variable. Expressed in algebraic form this becomes 

A 2 & 2 2 

EXERCISE 133 

.1. Express algebraically that the area of a circle 
varies directly as the square of the radius. 



How One Varying Quantity Depends on Another 327 

2. Assign at least 10 particular values to s in the 
formula for the area, A, of a square of side s, 
and compute the corresponding values of A. 
Tabulate these data. Then make a graph of 
this relation. Plot the values of s on the hori- 
zontal axis. 

3. The distance, d, which an object falls in any 
time, t, varies directly as the square of the time. • 
(t must be measured in seconds.) Make the 
algebraic statement for this law. 



4. Given that -J = -L fi n d j when ^,=400, 

d 2 t? 2 1 

t x = 5, and t 2 = 9. 

5. If the radius of one circle is twice that of 

another, then the area of one is ■ times 

the other. See Example 1. 

6. It is proved in geometry that the areas of two 
similar figures are directly proportional to the 
squares of any two corresponding sides. Using 
A x and A 2 as the areas of the similar triangles 
shown in page 82, and s t and s 2 as two corre- 
sponding sides, make an algebraic statement 
which will express the fact just stated. 

7. Referring to the general principle stated in 
example 6, we know that if a side of one tri- 
angle is twice its corresponding side in another 
triangle, then the area of the first triangle will 

be I times the area of the second triangle. 

How does A 1 compare with A 2 if s 1 = 2 and 
s 2 = 6 ? If s 1 = 5 and s 2 = 10 ? If s l = 4 and 
* 2 = 8? 



328 Fundamentals of High School Mathematics 

8. A baker sells two sizes of pies (the thickness is 
the same in both); one kind is 4 inches in di- 
ameter and the other is 8 inches. If the small 
ones are worth 10^ each, what are the large 
ones worth ? 

9. The volumes of two similar shaped objects are 
directly proportional to the cubes of any two 
corresponding dimensions, i.e. 

(a) A bucket 8 inches deep will hold ? 



times as much water as a similar one 4 
inches deep ? 

(b) A 10-inch cube of granite is how many 
times as large as a 5-inch cube ? 
, (c) Think of two spheres, one 4 inches in di- 
ameter, and the other 2 inches. The large 
one is I times as large as the small one ? 

(d) Find V 2 if V x = 125, d x = 5, and d 2 = 3. 

10. How much per dozen could you afford to pay 
for oranges 4 inches in diameter, if oranges 2 
inches in diameter sell for 25 fi per dozen? 
(Assume the same quality for both kinds.) 

11. If an iron ball 3 inches in diameter weighs 12 
pounds, how much will one 5 inches in diameter 
weigh ? 

12 Suppose it cost 20^ to buy enough leather to 
cover a 3-inch cube. What would enough 
leather to cover a 6-inch cube cost ? Explain. 

13. x varies directly as the square of y ; when x = 2, 
y =3. Find x when y = 5. 



How One Varying Quantity Depends on Another 329 

II. INVERSE VARIATION: THE STUDY OF VARIABLES 
WHICH ARE INVERSELY RELATED 

Section 152. When quantities are inversely related to each 
other. In the previous exercise the varying quantities 
were so related in any particular problem that an increase 
in one variable caused a corresponding increase in the other 
variable. Some variables, however, are so related that an 
increase in one is accompanied by a corresponding decrease 
in the other. 

An example : An increase in the rate at which a train 
moves causes a decrease in the time required to travel a 
certain distance. If the train travels at the rate of 20 miles 
per hour, it will require 5 hours to cover 100 miles ; but if 
it increases its rate to 30 miles per hour, it will decrease the 
time so that only 3^ hours will be required to make the 
trip. 

Let us illustrate this fact more in detail by tabulating 
the relation between the rate and the time of a train which 
makes a trip of 100 miles. Note from the table how a 
change in one variable, say the rate, is accompanied by a 
change in the other variable, the time. 

Table 24 



If the rate is 


10 


12§ 


15 


20 


25 


30 


33| 


40 


50 


then the time is 


10 


8 


6f 


5 


4 


3| 


3 


2| 


2 



This shows that an increase in the rate is accompanied 
by a decrease in the time. If we select any two values of 
the rate, say 20 and 50, and the corresponding values of the 
time, 5 and 2, we see that the ratio of the two values of the 



33° Fundamentals of High School Mathematics 

rate §-[}- is not equal to the ratio of the corresponding values 
of the time |. Clearly, |$ does not equal f, or, to use the 
more general form, 

r t 

-1 does not equal ^. 

These ratios would be equal, however, if we should invert 
one of them, e.g. 

20_2 r A _t A ^ 

50 ~ 5 ° r r^T^ 

The fact that the ratio of any two values of one of the va- 
riables is equal to the inverted ratio of the corresponding 
values of the other variable leads us to say that one of them 
is inversely proportional to the other, or varies inversely as 
the other. 

This gives the following principle : 

One variable is inversely proportional to another when 
the ratio of any two values of one of them is equal to the 
INVERTED RATIO of the two corresponding values of the 
other. 

EXERCISE 134 

1. The area of a rectangle is 200 sq. ft. Give 
several pairs of numbers that might represent 
its base and altitude. Then show that the ratio 
of any two particular values of the base is equal 
to the inverted ratio of the corresponding values 
of the altitude. In other language show that 

2. Ten men can do a piece of work in 32 days. 
Would an increase in the number of men cause 
an increase in the number of days ? If m rep- . 



How One Varying Quantity Depends on Another 331 

resents the number of men, and d the number 
of days, does d vary as m varies ? does d in- 
crease as m increases ? Suppose that 10 men 
could do the work in 32 days, or 20 men could 
do it in 16 days. From this fact, could we say 

that^ = -^l? Why not? Or that ^1 = ^2? 
d 2 m 2 d 2 m ± 

This study shows that the number of days re- 
quired to do a piece of work is — proportional 
to the number of men employed. 

3. Two variables, x and y, are inversely propor- 
tional ; V. e. ^1 = ^2. Find x if x = 12.4, j = 8.2, 

and y = 6.2. 

4. The time required to make a certain trip is in- 

t r 
versely proportional to the rate ; or, - 1 = — 2 - If 

the rate is 20 miles per hour, the time will be 
8 hours. What is the time required if the rate 
is 15 miles per hour ? 

Tell what kind of variation or proportionality is repre- 
sented by each of the following variables : 

5. The amount of flour which you can buy for a 
dollar, and the price per pound. 

6. The size of a 10 ^ loaf of bread, and the price 
of flour. 

7. The distance you travel on a railroad, and the 
carfare paid. 

8. The value of a fraction with a constant denom- 
inator, and its variable numerator. 



332 Fundamentals of High School Mathematics 

9. The value of a fraction with a constant numer- 
ator, and its variable denominator. 

10. The time it takes a boy to run a 100-yard 
dash, and his rate. 

Section 153. Graphical method of representing inverse 
variation. Figures 138 and 139 show graphically the re- 
lation between two numbers which are inversely propor- 
tional, or which vary inversely. It represents the base and 
altitude of a rectangle whose area is always constant, say 
100 sq. ft. 




Fig. 138 



This graph shows a series of rectangles of constant 
area, but with variable dimensions. It is very impor- 
tant to note that as the bases increase, the altitudes 
decrease. 

A more frequently used method of the graphic repre- 
sentation of this inverse variation is illustrated in the next 
figure. 



How One Varying Quantity Depends on Another 333 



8 

10 

CO 
CM 



! 






























1 
• 

1 






























1 

• 






























- 
































1 ■ 
\ 






























\ 






























\ 

1 


- 






























\ 
































■\. 


^.. 





















































4- 8 12 16 20 24- 28 32 36 40 44 48 52 

BASE 

Fig. 139. The line shows the relationship between two numbers 
which vary inversely ; in this case the relationship between the 
altitude and base of a rectangle whose area is constant, say 100 sq. ft. 
As the altitude increases, the base decreases. 



To construct this graph, the following table was made 
Table 25 



If base is 


2 


4 


5 


6 


8 


10 


12.5 


20 


then altitude is 


50 


25 


20 


16.6 


12.5 


10 


8 


5 



Note that as the base increases, the altitude decreases. 
How does the graph show this relation ? In what way- 
does this graph differ from those you have previously 
dealt with ? 



334 Fundamentals of High School Mathematics 
Show that the equation 

describes the relation between the base and altitude of any 
rectangle whose area is constant, say 100 sq. ft. 

EXERCISE 135 
GRAPHICAL REPRESENTATION OF INVERSE VARIATION 

1. The product of two variables, x and y, is always 
200. Tabulate 10 pairs of values of these vari- 
ables, and from the table construct a graph show- 
ing the way in which the variables are related. 
Measure values of x along the horizontal axis. 

2. Some tourists decide to make a trip of 100 miles. 
Show graphically the relation between (1) the , 
different rates at which they might travel, and 
(2) the time required at each rate. 

3. Think of some good illustration of inverse variation. 
Represent it graphically. 

REVIEW EXERCISE 136 

1. If 60 cu. in. of gold weighs 42 lb., how much 
will 35 cu. in. weigh ? 

2. If a section of a steel beam 10 yd. long weighs 
840 lb., how long is a piece of the same material 
which weighs 1250 lb. ? 

3. At 40 lb. pressure per square inch, a given pipe 
discharges 160 gal. per minute. How many 
gallons per minute would be discharged at 
65 lb. pressure? 



How One Varying Quantity Depends on Another 335 

4. A steam shovel can handle 900 cu. yd. of earth 
in 7 hr. At the same rate how many cubic 
yards can be handled in 5 hr. ? 

5. A train traveling at the rate of 50 miles per 
hour covers a trip in 5 hours. How long would 
it take to cover the same distance if it traveled 
at the rate of 35 miles per hour ? 

6. If 50 men can build a boat in 20 days, how long 
would it take 30 men to build it ? 

7. A wheel 28 in. in diameter makes 42 revolutions 
in going a given distance. How many revolu- 
tions would a 48-inch wheel make in going the 
same distance ? 

8. The volume, v, of a gas is inversely proportional 
to its pressure, p. Write an equation showing 
this fact. 

9. If the volume of a gas is 600 cubic centimeters 
(cc.) when the pressure is 60 grams per square 
centimeter, find the pressure when the volume 
is 150 cubic centimeters. 

10. When are two changing quantities or variables 
directly proportional ? When do they vary 
inversely ? 

11. How can you test for direct variation ? for in- 
verse variation ? Are x and y directly propor- 
tional in the equation x = 2 y + 5 ? 

12. In order to save d dollars in n years, how much 
would your savings have to average per month? 

13. The edge of one cube is x, and of another 2x. 
What is the ratio of their surfaces ? Of their 
volumes ? 



336 Fundamentals of High School Mathematics 

14. The radii of two spheres are r and 3 r. What 
is the ratio of their volumes ? 

15. The ratio of the diameter of the earth and the 
sun is -^-J-g-. The volume of the sun is how many 
times as large as the earth ? 

.16. The weight and distance (in a teeter board) are 
inversely proportional, i.e. 

W 2 d{ 

Find W 2 if W 1 = 80, d x = 10, and d 2 = 12. 

17. If it costs $ 75 to paint the exterior surface of a 
certain house, how much should it cost to paint 
a house whose dimensions are twice as great ? 
What principle of mathematics is involved here ? 



CHAPTER XVII 

SQUARE ROOTS AND RADICALS 

Section 154. Meaning of square root. In Chapter VI 
we found the square root of numbers which occurred in 
connection with the sides of a right triangle. At that 
time we did not make a special study of square root, but 
merely referred to the arithmetical method. This chapter 
will give a clearer notion of square root. 

The square root of a number is defined as one of its two 
equal factors. To illustrate, we might factor 100 in the 
following ways : 

4x25= 100 
5 x20 = 100 
10 x 10 = 100 
8 x 12.5 = 100, etc. 

Thus, by finding two equal factors, we see that the square 
root of 100 is 10. We shall use this definition constantly: 
the square root of a number is one of its two equal factors. 

In the same way the cube root of a number is one of 
the three equal factors of the number. Thus, — 2 is the 
cube root of - 8 because ( - 2)( - 2)( - 2) = - 8. 

Section 155. A number has two square roots. Just as 
10 was a square root of 100 because 10 x 10 = 100, so 
— 10 is also a square root of 100, because (— 10)(— 10} 
= 100. In the same way, either + a or — a is the square 
root of a 2 , because (a){a) = a 2 and (— a)(— a)= a 2 . 

Section 156. How to indicate the root of a number. It 
has been agreed to indicate the root of a number by the 
symbol V~ , called a radical sign. To designate what 
root is meant, a small number called an index is placed 
in the radical sign. Thus, V16 means the fourth root of 
16, i.e. 2, because 2 . 2 . 2 • 2 = 16 ; -\/27 means the cube 

337 



33 8 Fundamentals of High School Mathematics 

root of 27, i.e. 3, because 3 • 3 • 3 = 27 ; and V 25 means 
the square root of 25. It is customary, however, to omit 
the index when square root is meant. Thus, V25, without 
an index, is always understood to mean the square root 
of 25. 

EXERCISE 137 

FIND, BY TRIAL, THE ROOTS, WHICH ARE INDICATED, OF THE 
FOLLOWING EXPRESSIONS 



1. V9 

2. VP 



io. V& 



11. 



3. V25# 2 

4. </S 



5. V27^3 

6. </W 



7. V64* 8 



s. viooy 

9. VI 



fee 



49/ 



12. </,\ 



13. Vl44tf 1( > 



14. V400 xY 

15. -v/125 Pc* 

16. </16V 8 



17. V243J/ 10 

18. </2Mx^ 



Section 157. How to find the square root of algebraic 
expressions. We have seen that {a + bf or {a 4- b)(a -\-'b) 
= a 2 +2ab + ^ 2 . From this it is evident that the square 
root of a 2 + 2ab + b 2 must be a 4- b, or V# 2 + 2 ab + £ 2 
= (# + b). In the same way V;r 2 4- 10^r+ 25 is ^ + 5, be- 
cause (;tr 4- 5)(;r 4- 5) gives ^r 2 4-10^r+25. From, these 
illustrations we see that it is possible to extract the 
square root of an algebraic expression if we can show 
that it can be obtained by squaring some other expression ; 
that is, if we can show that it is the product of two equal 
factors. 



Square Roots and Radicals 



339 



EXERCISE 138 



Find the square root of the following expressions, where 
it is possible to do so. Check each. 



1. 


x 2 + 2xy+y 2 


8. 


y + 6^ + 20 


2. 


a 2 + 6a + 9 


9. 


25^ + 40^ + 16 


3. 


ft-±b+± 


10. 


^- 2 + 16 


4. 


/ 2_ 1 o 2f+ 25 


11. 


j/2_49 


5. 


4 a 2 + 12 a + 9 


12. 


12^+36+^r 2 


6, 


16+4r 2 + 8^ 


13. 


; 2 + ?i 2 + 2 fe 


7. 


1 + 21* + 100^ 2 


14. 


^4 _ 6 rs 2 + 9 



If any of the expressions above are not perfect squares, 
make the necessary changes to transform them into perfect 
squares. 

This chapter teaches how to find the square root of 
only one kind of algebraic expression, namely, an expres- 
sion which can be shown to be the product of two equal 
factors, or the square of an expression. 

Thus, to find the square root of an algebraic expression, 
you must show that it has been made by squaring another 
expression. If it is the product of two unequal factors, 
then we cannot, in this course, find the square root. 

EXERCISE 139 

FURTHER PRACTICE IN FINDING THE SQUARE ROOT OF ALGEBRAIC 
EXPRESSIONS 



1. *2_8*+16 

2. /2+12/+ 36 

3. 9 x 2 + 30 x + 25 

4. 4j/ 2 -20j/ + 25 

5. / 2 +/ + i 



6. . 2 + f, + i 

7. 25 b 2 +5£+i 

8 . 16 y _ lOj + l 

9. 8x + x 2 + 15 
10. •* - 6 rh 2 + 9 s* 



34-0 Fundamentals of High School Mathematics 

11. X 2 -\x+\% 15> y_2^ + i 

12. ^ 2 + f^ + T 9 6 16. 4 a 2 + 25 

13. y+Jj + J 17. l+9jr 2 + 6^ 

14. ;r 2 +;tr + 4 18. l+£2 + £4 

Section 158. Equations solved by finding square roots. 

We have seen that (+3)(+3)=9, and also (-3) (-3) 
= 9, and hence that 9 has two square roots, + 3 and 
— 3. In solving the equation 

* 2 = 9 
we are to find a number, or numbers, the square of which is 9. 
Evidently we find the square root of each side of the equa- 
tion, giving x = + 3 or — 3. (This is commonly written ± 3.) 

EXERCISE 140 

Find two values of the unknown which will satisfy each 
of the following equations. 

1. ;tr 2 = 16 9. x 2 - 49 = 

2. x 2 .= 64 10. y 2 - 25 a e b* = 

3. x 2 - 100 = 11. x 2 = (a + bf 

4. x 2 = 4 a 2 12. y 2 = c 2 + 2 cd + d 2 

5. y 2 = 81 bh 2 13. x 2 = 100O - b) 2 

6. x 2 = 2 2 a 2 14. -r 2 =2T 

7. x 2 = a 2 + 6 a + 9 15. x 2 - b 2 = 

8. j/ 2 = 4 £ 2 + 4 £ + 1 16. x 2 = 1 • 49 

Section 159. Square root of fractions. From the fact 
that (§ ) 2 — f * f = -|, we see that the square root of ^ must 

2 

be f . In the same way, the square root of — must be j , 

fa\ 2 a 2 
because (— J =t^- This suggests the principle that 



Square Roots and Radicals 



34i 



The square root of a fraction equals the square root 
of the numerator divided by the square root of the 
denominator. 

If the numerator or denominator of the fraction is not 
a perfect square, the approximate square root of the frac- 
tion is found in the same way, but the process is very long 
and unnecessarily laborious. To illustrate, the square root 
of -| by this method gives 

V 2 = 1.414 
V5 2.236 



^5 



.632. 



EXERCISE 141 

Find the square root of f ; of f ; of J. 

Section 160. Easier methods of finding the square root of 
a fraction. Consider the same example given in Section 
159 : V-f. Either of two methods will be much shorter 
than the previous method. 

1. Reducing the common fraction to a decimal fraction. 

|=V3"=.632. 
This calls for the square root of only one number, a 
decimal. 

2. Making denominator a perfect square. 

_ VlO 3.162 
25 



^ 12 _■ /10 



= .632. 



5 5 

The advantage of the second method is that it makes you 
find the square root of only one number, which is an in- 
teger rather than a decimal. 

This method is based upon the principle that both nu- 
merator and denominator may be multiplied by the same 
number without changing the value of the fraction. You 



342 Fundamentals of High School Mathematics 

must always multiply by a number that will make the de- 
nominator a perfect square. Why ? 

Which is the simpler form for computation ? 



oo V! ° r \l ° r 



( c ) V7 or 



21 

49 



V6 ? 
3 ' 

V21 



7. 

u4 



fl\ «/8 J 15 ^ 15 > 



3 ° r Vg ° r T" ? 



It should now be clear that the best method of finding 
the square root of a fraction is that of : 

1. Multiplying numerator and denominator by a 
number which will make the denominator a per- 
fect square ; 

2. Then finding the approximate square root of the 
numerator and dividing the result by the square 
root of the denominator. 



EXERCISE 142 
PRACTICE IN FINDING THE SQUARE ROOT OF FRACTIONS 

Compute to three decimal places. 



1. 


Vf 


6. 


v? 


11. 


Vf 


16. 


VA 


2. 


Vf 


7. 


V| 


12. 


VI 


17. 


Vf 


3. 


vj 


8. 


V? 


13. 


v? 


18. 


Vf 


4. 


V| 


9. 


Vf 


14. 


vf 


19. 


, / 2 
V 11 


5. 


Yf 


10. 


Vf 


15. 


vv- 


20. 


VA 



21. Find the square root, to three decimal places, 
of 2, 3, and 5. Fix these definitely in your 
memory. They will serve you well later. 



Square Roots and Radicals 



343 



Section 161. The square root of a product, one factor 
of which is a perfect square. The labor of finding square 
roots is greatly reduced by the use of a principle which is 
illustrated here. 

O) V12 = V4T3 = V3. V3 = 2 (1.732) =3.464. 

(£) V75= V25 - 3=V^5. V3 = 5(1.732) = 8.660. 
(c) V^b = V^> V£ = tfV£. 

Note here that the number whose square root is desired 
is the product of two factors, one of which is a perfect 
square. The principle used here is usually stated : 

The square root of a product is equal to the product 
of the square roots of the factors, or 

Va6 = Va • V&. 

In applying this principle, note that the number whose 
square root is desired must be separated into two factors, 
one of which is a perfect square. The square root of 
this factor must then be multiplied by the approximate 
square of the other factor. For example : 

(a) V200 = V100T2 = VTOO • V2 = 10(1.414) = 1.414. 
(&) V20 = V3T5 = V4 • V5 = 2V5 = 2(2.226) = 4.452. 
Those students who know the square root of 2, 3, and 
5 will have a great advantage in this work. 



EXERCISE 143 
Solve by the short method. 

1. Vl2 4. V108 7. V32 10. V72 

2. V75 5. V27 8. V18 11. V20 

3. V300 6. V50 9. V8 12. V45 



344 Fundamentals of High School Mathematics 

13. V80 15. VI25 17. V288 19. VI50 

14. V500 16. VS8 18. V147 20. V54 

21. Use this method of square root in rinding the diago- 
nal of a square when each side is 10 ; 5 ; 8 ; 20 ; 12. 



RADICALS 

Section 162. Definition of radical. An indicated root of 
any expression is called a radical. The indicated square 
root of 10, i.e. VlO, is a radical. Similarly V150, Vx, 
Vr 2 + 10 x + 25, -\/8, etc., are radicals. 

Section 163. Radicals may be expressed in different 
forms. The form of a radical expression may be changed 
without changing its numerical value. Recall that in 
Section 161 we changed the form of radicals to put them 
in a more convenient form for the computation of square 
roots ; for example, V50 = V25-2 = 5 V2. Here we have 
three different forms, each having the same numerical 
value. Also, recall that we changed Vf to V||- and then 

to • , giving us three different forms of the same radical. 
5 

Section 164. Radicals should be reduced to the simplest 
form for computation. By comparing the following radi- 
cal forms you will be able to state when a radical is in its 
simplest or most convenient form for computation. 

Which is most easily computed : 

(a) V50 or Vl(T5 or V25T2 or 5 V2 ? 

(b) V32 or V84 or V16T2 or 4 V2 ? 






VI 

or or 

V2 



V3 
- or — = or 

5 V5 



J 2 V2 > 



15 V15, 
25 ° r — ? 



Square Roots and Radicals 



345 



These examples suggest the following important prin- 
ciple concerning the simplest form of radicals : 

A radical is in its simplest form when it contains no 
factor of the same power as the root indicated, and 
when no radical occurs in a denominator. 

fa 
For example, -y- is not in its simplest form, for the radical 

V^ occurs in the denominator. This is reduced to the' 
simplest form by multiplying both numerator and denom- 
inator by an expression which will make the denominator a 
perfect square. Thus 

la . lad "\f~ab 



^~b ^b 2 b 

The last radical is in its simplest form. Also, the radical 
^/a 2 b is not in its simplest form, because the factor a 2 is 
of the same power as the root. To reduce it to the 



simplest form we must take the square root of the 



a- 



•y a 2 b — a~Vb. The last radical 



is in its 



which gives 
simplest form. 

Which of the following radicals are in their simplest 
form : 



4 IT' ¥' 4 ^f IP iVB > V *' V ^' VT 



vio, 

12, V20, Va*d, VaT 2 , Va~b, 2V8, «VS, and bVab. 



EXERCISE 144 
PRACTICE IN CHANGING RADICALS TO THEIR SIMPLEST FORM 

Do not find numerical values. 
l. V8 3. Vl2 5. V200 7. V^ 



2. 



vio 



4. V20 



6. ^/ab 8. V25^/ 



346 Fundamentals of High School Mathematics 



9. V8tf£ !2. V| 

10. ^Wxy 13. V| 



15. 



a 



11. VJ 



14. VJ, 



— 16 



■4 






19. The base and altitude of the right triangle shown 



here are R and — - • Express in sim- r 



b^ 



plest radical form the hypotenuse. R 

20. Find the simplest radical expression for the 
diagonal of a rectangle whose dimensions are 
n and 2 n ; n and 3 n ; 3 /z and 4 ;z. 

21. What is the diagonal of a rectangle whose 
dimensions are \ s and j ? %s and 2 j- ? -| ^ 
and s ? 



TIMED PRACTICE EXERCISE J 

How many examples in this exercise can you do in 3 
minutes? The very rapid and accurate pupils in 100 cities 
did 16; the average about 10. 



1. V8 


8. 


V^iy 


15. 


v? 


2. Vtf 3 £ 4 


9. 


V2 
V 5 


16. 


V18 


3. V| 


10. 


V50 


17. 


■y/rs 1 


4. V27 


11. 


Vtf 5 £ 4 


18. 


vs 


5. V^ 6 


12. 


V| 


19. 


V24 


*'V£ 


13. 


V45 


20. 


VF?° 


7. V12 


14. 


V/z 5 / 6 


21. 


Vf 



Square Roots and Radicals 



347 



H 



REVIEW EXERCISE 145 

1. The area of a circle is expressed by the for- 
mula, A=ttR 2 . (tt = 3.1416.) Solve for R. 
Find R when ,4 = 3.1416; when ,4 = 100; 
when A = ir ; A = 500. . 

2. The entire area of the cylinder 
shown here is 2 ttR 2 + 2 irRH, or 
2 ttR(R + 7r). Find the entire area 
when R = 3.5. "~^R 

3. The area of a circle whose radius is 10 inches 
is 314.16 sq. in. What is the area of a circle 
whose radius is twice as great ? 

4. The sides of a triangle are 12, 28, and 32. The 
perimeter of a similar triangle is -| that of the 
given triangle. Find the sides of the second 
triangle. 

5. The legs, altitude, and base of a right triangle 
are equal. Find each if the hypotenuse is 30. 

6. The legs of a right triangle are —- and — . 
Find the hypotenuse. 

7. The area of a right triangle is 50. If the base 
and altitude are equal, what is the length of 
each ? 

8. One leg of a right triangle is — and the hy- 
potenuse is R. Find the other leg. 

9. What is the tangent of one of the acute angles 
of a right triangle, if the legs are equal ? Make 
a drawing. 



348 Fundamentals of High School Mathematics 



10. What is the cosine of one of the acute angles of a 
right triangle whose base and altitude are equal ? 

Draw an equilateral triangle of side s. Solve 
for the altitude and for the area. 

Find the altitude of an equilateral triangle whose 
sides are 10 inches. What is its area ? 

One angle of a right triangle is 30°. The hy- 
potenuse is 20. Find the other two sides. 

One angle of a right triangle is 30°. The hy- 
potenuse is 20. Find the other two sides. 
Compare this example with the previous one. 

The side of a square room is 21.5 feet. Find 
its diagonal correct to two decimals. 

What is the perimeter of a square whose diag- 
onal is 12 inches ? 

The side of a square is a. What represents its 
area ? its perimeter ? its diagonal ? 

A rectangle is four times as long as it is wide. 
Find its diagonal if its area is 576 square inches. 
Figure 141 is an equi- 
lateral triangle, and CD 
is perpendicular to AB. 
Find CD if each side 
of the triangle is 20 
inches. Then find the 
area of triangle ABC. 
The area of a right 
triangle is 24 square 
inches. Its base is 6 inches, 
and hypotenuse. 



11. 



12. 



13 



14. 



15. 



16 



17 



18 



19. 



20. 




Fig. 140 
Find its altitude 



Square Roots and Radicals 



349 




Fig. 141 



21. The diagonal of a square is d. Show that 
(side) is - — -■ 

22. CD, the altitude of 
equilateral triangle 
ABC, is 16 inches. 
Find the sides of the 
triangle and its area. 

23. How long an umbrella 
will lie flat down on the 
bottom of a trunk whose inside dimensions are 
27 inches by 39 inches ? 

24. Can a circular wheel 8 feet in diameter be taken 
into a shop if the shop door is 4i feet wide and 
6J feet high ? 

25. If A represents the area of a square, what will 
represent its perimeter ? its diagonal ? 

26. The sides of a triangle are 12, 16, and 24 inches. 
Is it a right triangle ? Why ? 

27. If you know two sides of a triangle, can you 
always find its area? Explain. 

28. Find the area of a square whose diagonal is 12 
inches longer than one of its sides. 

29. Will an umbrella 30 inches long lie flat down in 
a suit case whose inside dimensions are 18 by 
25 inches ? 

30. A rectangle is 12 by 18 inches. How much 
must be added to its length to increase its diag- 
onal 4 inches? 



35° Fundamentals of High School Mathematics 

31. One side of a right triangle is 3 times as long 
as the other. The hypotenuse contains 30 
inches. Find the area of the triangle. 

32. Find the diagonal of a square whose side is 20. 

33. Make a formula for finding the diagonal of any- 
square. 

34. Find the sides of a square whose diagonal is 
20. 

35. Make a formula for finding the side of any 
square whose diagonal is known. 

36. When is a radical in its simplest form ? 

37. Factor 12, r 2 - 2x -24. 38. )2.r=4;r+6 

39. It costs the same to sod a square piece of ground 
at 15 ^ per square yard as to put a fence around it at 60 ^ 
per yard. Find the side of the square. 

40. In how many years will $ 300 double itself at 6 % 
simple interest ? 



CHAPTER XVIII 

HOW TO SOLVE EQUATIONS OF THE SECOND DEGREE 

Section 165. What are quadratic equations? In all 

the previous chapters you have solved equations of the 
first degree ; that is, equations in which the unknown (or 
unknowns) did not have exponents greater than 1. This 
chapter will show how to solve equations of the second de- 
gree, equations in which the unknown occurs to the second 
power. To illustrate, you will learn how to solve equations 
such as 

„r 2 + 8;tr=20. 

The fact that the unknown, x, in this equation occurs in the 
second power or second degree (as, x 2 ) leads us to speak of 
the equation as a second-degree, or quadratic, equation. 

Three ways of solving second-degree or quadratic equa- 
tions will be explained. These, in the order in which we 
shall discuss them, are : 

I. Solution by factoring. 
II. Solution by completing the square. 
III. Solution by graphical representation. 

Section 166. Our real problem in solving a quadratic 
equation is to find a value of x which will make the left 
side equal to the right side. To solve the quadratic equa- 
tion x 2 - 8 x + 12 = 

we are trying to find a value which will make the left side, 
x 2 — Sx -\- 12, equal to the right side, 0. It helps to think 
of a quadratic equation (just as we thought of a simple 
equation in Chapter I) as asking a question : what value, or 
values, can x have, to make the expression x 2 — Sx + 12 
equal to zero ? 

To solve the quadratic equation x 2 — 8 x -j- 12 = 0, we 
have to make a special study of the expression x 2 — 8 x+ 12, 

351 



352 Fundamentals of High School Mathematics 

to find some value for x which will make this expression 
equal to zero. It will help us to recognize that the value 
of this expression depends upon what value we give to x. 
If jt is 1, the value of the expression is 5 ; if x is 2, the 
value of the expression is 0. Thus we have here two va- 
riable quantities : x itself is one variable, and the value of 
the expression x 2 — Sx + 12 is the other variable. 

Section 167. The trial method : laborious and not sure. 
One way to find the desired value of x is to assign various 
values to x in the expression x 2 — &x + 12, hoping to find 
a value for x which will make x 2 — Sx + 12 = 0. In doing 
this, one class made the tabulation given in Table 26. 

Table 26 



If x is 





1 


2 


3 


4 


5 


6 


7 


-1 


-2 


thenx 2 -8x+12is 


12 


5 





-3 


-4 


-3 





5 2132 



From this table, do you see any value of x which makes 
the expression x 2 — 8x + 12 equal to zero? If so, that 
value of x satisfies the quadratic equation x 2 — 8x+ 12 = 0. 
How many values of x will make the expression zero ? 
Check them to be certain. 

This method is very laborious, and might often fail to 
give the value of the unknown. It just happens to give 
two values here, but it might more easily not happen to 
give them. It is merely a trial method. 



I. HOW TO SOLVE QUADRATIC EQUATIONS BY 
FACTORING 

Section 168. An economical method : factoring, to get a 
product equal to zero. The use of a principle which we 
already know gives an easy method of solving quadratic 



Solving Equations of the Second Degree 353 

equations. The principle is : a product of any number of 
factors is zero, if one of the factors is zero. Recall that 
4-5-0 = 0, or that a • b • c == if either a y b, or c is 0. 
Why ? Because any number multiplied by is 0. Under 
what conditions is the product xyz zero ? Evidently if 
either x, y, or z is zero ; that is, if one of the factors is 0. 

In the same way, the expression (x — 6)(x — 2) which is 
the product of two factors, could be if x — 6 were 0, or 
x— 2 were 0. 

Illustrative example. Now let us apply this principle to 
the solution of the quadratic equation x 2 — 8x -f- 12 = 0. 

Solution: x 2 - 8* + 12 = 0. 

In order to solve the equation, we want to get a product 
equal to zero. Hence we should factor the expression 

Forming a product (i. e. factoring) gives 
(x - 2)0 - 6) = 0. 
Now we have a product equal to zero. But in order to 
have a product equal to zero, one of its factors must be 
zero. 

Therefore, if x — 2 = 0, x must equal 2. 

Similarly, if x— 6 = 0, x must equal 6. 

This method gives as the values of x, 2 and 6, much 
more easily than by assigning values to x as in Table 26. 
A second illustrative example will enable you to use this, 
method. 

Illustrative example. 

Solve the quadratic equation, 

x 2 + 2 x = 48. 
Solution : x 2 + 2 x = 48. 

We want a product equal to ; therefore the equation should be 
in the form of an expression equal to 0, 
or jc 2 + 2 x — 48 = 0. 



354 Fundamentals of High School Mathematics 



(1) Forming a product, 

(x + 8)(*-6)=0. 

(2) Making the first factor = 0, 

X + 8 = 0, 
or x = — 8. 

(3) Making the second factor 0, 

x - 6 = 0, 
or x = 6. 

Check : 64 - 16 = 48. 

36 + 12 = 48. 



EXERCISE 146 
PRACTICE IN SOLVING QUADRATIC EQUATIONS BY FACTORING 

1. What values of x will make x 1 — 5x + 6 equal 
to ? or in other words, solve the equation 

2. For what values of y does y 2 — 7jj/ + 14 = ? 

3. What values of x will satisfy the equation 

^2 + ^-20 = 0? 

Solve each of the following equations : 

4. a a + -9d!-22 = 

5. x 2 + lx=l$ 

6. y 2 -y = 20 



12. J^ 2 +^-= 12 
13. 



2 ^4 



7. £2_ 36== 

8. ^2_ 8r = _ 16 

9. m 2 == ;« + 2 

10. ^_,_^ = 5 6 

11. 2j/ 2 + 5j/-12 = 



14. ^ 2 + 2^=0 

15. 12=^2 + ^ 

16. O+6)O-2)=:0 

17. 3£ 2 + 7£ = _2 

18. 2^ 2 + Sx- 3 = 



19. One number is 2 larger than another. Their 
product is 80. Find each number. 

20. The sum of two numbers is 10 ; the sum of 
their squares is 52. Find each number. 



Solving Equations of the Second Degree 355 

21. The altitude of a triangle is 4 inches longer 
than its base. The area is 96 square inches. 
Find the base and altitude. 

22. The square of a number is 30 more than the 
number itself. Find the number. 

23. The perimeter of a rectangle is 60 inches, and 
its area is 200 square inches. What are its di- 
mensions ? 

24. A boy covered a cubical box with paper. He • 
needed 96 sq. in. What is the length of one 
edge of the cube ? 

25. The sum of the squares of three consecutive 
numbers is 50. What are the numbers ? 

26. Find the length of each side 
of the triangle shown here. 

27. A rectangular floor is 6 ft. x+ ' 7 
longer than it is wide ; its area is 34 sq. yd. 
What is its length ? 

28. Determine the area of the triangle in example 
26. 

29. 4;r 2 = 81 31. jj/ 2 - 6 by -f 9 b 2 = 

30. x 2 -9a 2 = 32. x 2 = 8ax-lQa 2 

II. HOW TO SOLVE QUADRATIC EQUATIONS BY 
COMPLETING THE SQUARE 

Section 169. The left side of the equation must be a 
perfect square. By this method of solving quadratic 
equations we find the square root of each side of the equa- 
tion. Thus to solve the equation 

x 2 + 6 x = 27 




356 Fundamentals of High School Mathematics 

we find the square root of each side. But x 2 -f 6 x is not 
a perfect square; we cannot extract the square of an 
algebraic expression which is not a perfect square (the 
approximate square root of an arithmetical number may- 
be found, but not of an algebraic expression). Therefore, 
to use this method, we must make the left side of the 
equation a perfect square, or " complete the square." An 
exercise will recall how to do this. At this point, the 
examples on pages 268 and 269 should be reviewed. 



EXERCISE 147. ORAL WORK 



PRACTICE IN MAKING PERFECT SQUARES, BY ADDING CERTAIN TERMS 
TO GIVEN EXPRESSIONS 

1. What is the square of x + 3 ? 

2. What is the square of x + 5 ? 

3. What does x 2 + 6x lack being a perfect square ? 

4. What does y 2 + 10j/ lack being a perfect square ? 

What should be added in order to make each of the 
following expressions perfect squares ? 

5. x 2 4- 4 x + ? 



6. x 2 - 6 x + ? 

7. x 2 + lSx-{- ? 

8. j/ 2 + 5j + ? 

9. p 2 + 3/ + ? 
10. x 2 + x + ? 



11. x 2 - 7 x + ? 

12. x 2 — x + ? 

13. x 2 + \x + ? 

14. x 2 +\x+ ? 

15. x 2 - | x + ? 

16. ^2_| ;r + ? 



The solution of these examples will suggest the follow- 
ing method for completing the square of an expression of 
the type x 2 + px: i.e. 



Solving Equations of the Second Degree 357 

Add to the expression the square of ^ of the coefficient 
of x, i.e. (|) 2 . 

Illustrative example, continued. We are now able to solve the 
equation x 2 + 6 x = 27. 

(1) Adding 9 to each side, to " complete the square," 

x 2 + 6 x + 9 = 36. (Why 9 ?) 

(2) Extracting the square root of each side gives 

x + 3= + 6or-6. 
Using +6, x = 3. 

Using - 6, x = - 9. 

Check : 9 + 18 = 27. 

81 - 54 = 27. 

EXERCISE 148 

Solve these quadratic equations by " completing the 
squares." 

6. y 2 + 1 y = 8 

7. x 2 + x = 56 

8. c 2 + 6 c = 11 
16 9. p 2 - 4/ = 36 

10. lQx+ x 2 = - 9 

11. For what value of x does x 2 + 6 x = 36 ? Could 
this equation be solved by the factoring method ? 

12. Does 8 x ever equal 9 — x 2 ? 

13. A rectangular field is 2 rods longer than it 
is wide, and it contains 6 acres. Find its 
width. 

14. Find two consecutive even numbers whose 
product is 48. 

15. x 2 - 4 x - 32 = 17. x 2 - 7 x - 18 = 

16. x 2 = 18 18. y + 6 = j/ 2 



1. 


^2 +6 ^ = 40 


2. 


*2-8*=84 


3. 


J/ 2 + 2j = 15 


4. 


^2 _ io^ = _ 


5. 


*2 + 3 / = 10 



358 Fundamentals of High School Mathematics 



19. p 2 - 5^ = 11 

20. x 2 — 2 x = f 

21. * 2 -f * = 20 



22. y +j/ = 30 

23. y 2 - 3j = 10 



24. ^ 



+ * = f 



25. Make a good rule for solving these equations. 

Section 170. In order to complete the square the equa- 
tion must be in such form that the coefficient of x 2 is 1. If 
the equation is given in such form as 2x 2 + 5;r=12, it must 
be changed so that the first term is x 2 . 

Illustrative example. 

Solve 2 jc 2 - 5 x = 12. 

(1) Dividing each side by 2, to make x 2 have 1 for its coefficient, 



x°~ 



x = 6. 



(2) Adding (£) 2 to each side, 



8 16 



1 — f 



Why 



(3) Extracting the square root of each side, 

x-f=V,or-y 

(4) Using + V-, x = -¥-> or 4. 

(5) Using - JjL, x = I - \S or - f. 

(6) Check : The pupil should check the example. 



EXERCISE 149 
Solve and check each of these equations: 
1. 2 x 2 + 5 x = 7 



2. 


3^ 2 + 4^r= 16 


3. 


2 j/ 2 +j/ = 19 


4. 


2^+* =10 


5. 


2j/ 2 -y = 6 


6. 


2y_lQ 7 = _8 


7. 


3;r 2 + 5;r = 8 


8. 


5^+*=! 



9. 


Zy 2 -y = i 


10. 


2x 2 -x-l=0 


11. 


3 £ 2 - 2 £ = 8 


12. 


4 a 2 + a = 5 


13. 


5y _ 10j/ = 15 


14. 


r 2 - J <: = 3 


15. 


f-\y = % 


16. 


2 / 2 -i / = 35 



Solving Equations of the Second Degree 359 

EXERCISE 150 

Solve and check. 

1. 2^ 2 + 10^=T2 5. ?£--*=3 

o 

2. 3tf 2 + 6a=45 6 . f-? = fl 

2 „r 10 

** ,*_ q „ ^ 2 3.r_31 

Hint : Get rid of fractions. 8 . 3 y 1 + 5 y = 22 

4. y-40 = 8y 9. 5^ 2 + 16^ + 3 = 

10. The difference of two numbers is 4, and the 
sum of their squares is 210. Find the numbers. 

11. A farmer has a square wheatfield containing 10 
acres. In harvesting the wheat, he cuts a strip of 
uniform width around the field. How wide a strip 
must be cut in order to have the wheat half cut ? 

12. Divide 20 into two parts whose product is 96. 

13. The sum of two numbers is 20, and the sum of 
their squares is 208. Find the numbers. 

14. I went to the grocery for oranges. The clerk 
said they had advanced 10 cents per dozen. I 
got J dozen fewer oranges for a dollar. What 
was the original price per dozen? 

15. A piece of tin in the form of 
a square is taken to make an 
open-top box. The box is 
made by cutting out a 3-inch 
square from each corner of 
the piece of tin and folding 
up the sides. Find the length 
of the side or the original 
piece of tin if the box contains 243 cubic inches. 





1 

1 





360 Fundamentals of High School Mathematics 



16. 



17. 



A rectangular park 56 rods long and 16 rods 
wide is surrounded by a boulevard of uniform 
width. Find the width of this street if it con- 
tains 4 acres. 

The members of a high-school class agreed to 
pay $8 for a sleigh ride. As 4 were obliged to 
be absent, the cost for each of the rest was 10 
cents greater than it otherwise would have been. 
How many intended to go on the sleigh ride ? 



III. HOW TO SOLVE QUADRATIC EQUATIONS 
GRAPHICALLY 

Section 171. Graphs show the relation between variables. 
We have frequently pointed out that graphs are used to 
represent relationship between related variables. Then, 
to solve a quadratic equation graphically, we must recog- 
nize these variables and plot specific values for them. 
Consider the equation : 

X*-%X+\2 = Q. 

The expression x 1 — 8;r + 12 may have many values, de- 
pending upon the value of x. Thus we have two related 
variables : x itself is one variable, and x 2 — 8 x -f 12 is the 
other one. To represent graphically the many values of 
x 2 —8x + 12, we must assign values to x } and compute 
the corresponding values of x 2 — 8x + 12, i.e. we must 
tabulate the related values of the two variables. 
Table 27 gives these values. 

Table 27 



If jxis 





1 


2 


3 


4 


5 


6 


7 


-1 


-2 


then x~8x+i2 is 


12 


5 


O 


-3 


-4 


-3 





5 


21 


32 



Solving Equations of the Second Degree 361 



Next, we plot the various values of these two variables. 
For convenience, we plot the unknown x, on the horizontal 
axis, and the expression 
x 2 — 8 x + 12 on the ver- 
tical axis. Locating the 
points represented by 
the pairs of values, (0, 
12), (1, 5), (2, 0), etc., 
from the table, we obtain 
as the graphic represen- 
tation of. x 2 — Sx + 12, £. 
the curve of Fig. 144. 
From this graph, what 
is the value of x 2 — 8 x 
+ 12 when;r= 2 ? when 
x = 4 ? when x = 6 ? 
when x= 7 ? ' Does the . 
graph show for what val- 
ues of x the expression equals ? If it does, then it solves 
the equation x 2 — 8 x + 12 = 0. 

This graph should emphasize the fact that in the 
quadratic equation x 2 — Sx + 12 = 0, the left side, 
x 2 — 8^+12, may have many values, depending upon 
the value of x, and that this expression is for two 
particular values of x; namely, when x=2 and when 
x=6. These are the points at which the curve cuts 
the *-axis. 

A study of the graphical solution of another quadratic 
equation, x 2 — 2x— 35 = 0, Fig. 145, will make this method 
of solving quadratics still clearer. 

By referring to it, you will be able to answer the fol- 
lowing questions. 



Fig. 144 



362 Fundamentals of High School Mathematics. 



Y 

















































































































































































































































I 




















/ 


























I 




















/ 














































/ 


























\ 




















1 


















1C 








-5 






r 


> 












f 


10 










15 














I 


















1 




























I 
















































| 
















/ 
















































































-■( 


) 










/ 














































i 






























\ 
















' 1 






























1 
















1 r* 






























\ 
















! , 


































-^( 


) 










1 I 
































I 












/ 


\' 
































\ 














tf 
































\ 












A 


/ 


































| 












r~ 


































-'f 


) 






/ 




V' 










































' 






































V 






/ 










































's 


t 




























- 


























































-/ 


■) 



























































































































Fig. 145 



1. How many different values may the expression 
x 1 — 2 x — 35 have ? 

2. When is the value of the expression zero ? i.e. 
where does the curve cross the ;r-axis ? 

3. Could the curve be drawn accurately by tabulat- 
ing only two or three pair oi values ? Why ? 

Summary of steps in the graphical solution of quadratic 
equations. A study of these illustrations suggests the 
following : 

1. Graph the quadratic expression which forms one 
member of the equation. (The other member 
should be zero.) 



Solving Equations of the Second Degree 363 

2. From the graph find for what values of x the 
expression is zero. In other words, find at what 
value of x the graph cuts the x-axis. These values 
are the values of x which will satisfy the equation. 

3. Check your result by substituting these values 
of the unknown in the original equation to see if 
they do satisfy it. 

EXERCISE 151 
PRACTICE IN SOLVING QUADRATIC EQUATIONS GRAPHICALLY 

Solve the following quadratic equations by drawing 
graphs of the expressions : 

1. x 2 -5x-U = 6. 2/ + 5j/ + 3 = 

2. ;r 2 + 3.r=40 7. ;r 2 + 8.r + 16 = 

3. y 2 - y = 20 8. y 2 + 3 y -f 1 = 

4. 2.r = 48-.r 2 9. (>- 2) 2 + 6;tr = 12 

5. ^ 2 -6,r+9 = 10. „r 2 + 4 = 



In these examples, did you find any graph that did not 
cut the .r-axis in two places ? What conclusion would 
you draw if the graph just touched the ;r-axis ? if it did 
not even touch it ? 

What seem to be the disadvantages of the graphical 
method of solving quadratic equations ? the advantages ? 

REVIEW EXERCISE 152 

1. By substituting any value for x in x 1 — 1, 2x, 
and x 2 4- 1, show that the three numbers which 
result are sides of a right triangle. 



364 Fundamentals of High School Mathematics 

2. If the sides of a right triangle are 6 inches and 

8 inches, then the hypotenuse must be • 

inches. 

3. How can you tell when a triangle is a right tri- 
angle without measuring its angle ? Is the 
triangle whose sides are 5, 12, and 13 a right 
triangle ? Why ? 

4. What is the area of an equilateral triangle each 
of whose sides is 30 inches ? 

5. How would you find the side of a square which 
had the same area as a circle with a radius of 
12 inches ? 

6. Write a formula for b if a> b, and c are the alti- 
tude, base, and hypotenuse of a right triangle ; 
similarly, a formula for a. 

7. Express the hypotenuse of a right triangle 
whose altitude exceeds its base by 6 inches. 

8. When is it impossible to find the square root of 
a number ? 

9. The sum of two numbers is -^ and their product 
is |. Find the numbers. 

10. The hypotenuse of a right triangle is 25 feet. 
Find the other two sides if you know that their 
sum is 35 feet. 

11. A piece of wire 30 inches long is bent into the 
form of a right triangle whose hypotenuse is 13 
inches. Find the other sides of the triangle. 

12. The area and the perimeter of a rectangle are 
each 25. What are its dimensions ? 



Solving Equations of the Second Degree 365 

13. A photograph, 8 inches by 10 inches, is enlarged 
until it covers twice the original area, keeping 
the ratio of the length to the width unchanged. 
Find the sides of the enlarged photograph. 

14. In placing telephone poles between two places, 
it was found that if the poles were set 10 feet 
farther apart than originally planned, 4 poles 
fewer per mile were needed. How far apart 
were the poles placed at first ? 

15. Solve the equation 

(2x+l)(x r 2)=(x + 2)(x-l)+5. 

16. The altitude of a right triangle exceeds the base 
by 7. The hypotenuse is 13. Find the base 
and altitude. 

17. Using the formula s = 16 1 2 for the distance, s, 
an object will fall in / seconds; determine how 
long it would take a stone to reach the ground 
if dropped from a height of one mile, (s is 
measured in feet.) 

18. Solve for x : x + 1 = 20 x* 1 . 

19. The area of the rectangular flag in a certain 
school is 98 sq. ft. and its perimeter is 42 ft. 
Solve for its dimensions. 

20. How much area will a piece of wire 100 ft. 
long inclose if bent in the shape of a square ? 
if bent in a circle ? 

21. Express the area formed by bending a piece 
of wire 10 ft. long into a square ; by bending it 
into a circle. 

22. Solve for v and R : F= ™—. 

R 



366 Fundamentals of High School Mathematics 

Mv 2 



23. Solve for M and v : E 



2 



24. From the formula 2fs = v 2 — u 2 , derive a for- 
mula for v. 

25. S—Vt— \gt 2 is a formula used by artillery 
officers. Derive a formula for V. 

26. The distance, d (in miles), which one can see 
out on the ocean from the height, h (in feet), is 
expressed by the formula d= 1.22 Vk. How- 
far could you see from a height of 50 feet ? 
How high would one have to be to see a dis- 
tance of 8 miles ? 

27. Two travelers leave a cross roads at the same 
time, one going due east 6 miles per hour, and 
the other due south 8 miles per hour. In how 
many hours will they be 40 miles apart ? 

28. A tourist has a trunk 32 inches long. It will 
just permit an umbrella 38 inches long to lie 
diagonally on the bottom. A trunk how much 
longer would be needed if he wanted to carry 
a gun 6 inches longer than the umbrella? 

29. How many rods of fence will be required to 
inclose a square field containing 40 acres ? 



INDEX 



Algebraic expression, meaning of, 39; 
numerical value of, 39. 

Algebraic fractions, defined, 285; how- 
to add or subtract, 285 ; how to mul- 
tiply, 292. 

Algebraic sum, 172. 

Angle, how determined, 55; how to 
describe, 6i ; tangent of an, 89, 90, 
96. 

Angles, how to measure, 57. 

Angular measurement, unit of, 56. 

Average, 139, 140. 

Bar graph, 117, 118, 121. 
Binomial, 266. 

Central tendency, 139. 

Collecting terms, 176. 

Common factor, 271, 272, 273, 278. 

Computation, rules of, 4. 

Constants, 152, 319. 

Corresponding angles, 84. 

Cosine, 102. 

Cosine of an angle, 100. 

Cosines and tangents, table of, 97. 

Degree, 56. 

Denominator, most convenient, 286. 

Direct variation, 321-329. 

Distance scale, 168. 

Division of fractions, 295. 

Equation, 13; use of, 22; is satisfied 

when, 28. 
Equational or formula method, 117, 148. 
Equations, how to check, 27 ; literal and 

fractional, 300; of second degree, 

351-366. 
Equivalent fractions, 282. 
Evaluation, 40. 
Exponents, use of, 41. 

Factor, 271. 

Factoring, cross-product method of, 273. 

Factors of an algebraic expression, 271. 

Finding unknowns, 108. 

Formula or algebraic method, 158. 

Formula method, 154, 155. 

Formulas, construction of, 44. 

Fraction, terms of, 282. 



Fractions, to get rid of, 30 ; with letters, 

282; principles in handling, 283; 

steps in adding or subtracting, 288; 

how to divide, 295; with binomial 

denominators, 306. 
Frequency table, 35, 137. 

General formulas, construction of, 310. 
Graphic method, 117, 148, 154, 155, 

158. 
Graphic representation, summary of, 

129. 

Hypotenuse Rule, 108, 109. 



Interpolate, 126. 
Inverse variation, 329-336; 
method, 332-335. 



graphical 



Letters to represent numbers, 1. 
Line graph, 124, 125, 126. 
Literal equations, 300. 

Mathematical law, 158. 
Mathematics, chief aim of, 147. 
Measurement of angles, 55. 
Median, 139, 140. 
Mode, 139. 

Most convenient denominator, 286. 
Most convenient multiplier, 30. 
Multiplication of fractions, 292. 

Negative numbers, 166. 

Parentheses, how used to indicate mul- 
tiplication, 182. 

Positive numbers, 166; and negative 
numbers, 169. 

Practice Exercises, 47, 176, 206, 254, 
269, 279, 280, 305, 309 ; how to use, 
47; Record Card, 49. 

Prime factors, 277, 278. 

Protractor, 57. 

Pythagorean Theorem, 110. 

Quadratic equations, 351-366; solved 
by factoring, 352-355; solved by 
completing the square, 355-360; 
solved graphically, 360-366. 



367 



3 68 



Index 



Radicals, 344-350. 

Radical sign, 337. 

Ranking, 135. 

Rank order, 135, 137. 

Ratio, 76, 89, 100. 

Relation, between quantities, 147; be- 
tween two variables, 154. 

Relationship, picture of, 150. 

Right triangle, use of, in finding un- 
knowns, 89. 

Rules of computation, 4. 

Satisfying an equation, 28. 

Scale drawings, 54. 

Signed numbers, 166, 169; how to com- 
bine, 172 ; how to multiply, 178. 

Similar figures, 75. 

Similar triangles to find unknown dis- 
tances, 75. 

Solving an equation, 13. 

Special product, 266. 

Square root, 337-340 ; as in arithmetic, 
110; and radicals, 337-350 ; of alge- 



braic expressions, 338-340; of frac- 
tions, 340. 

Statistical graph, 147, 158. 

Statistical table, how to prepare, 134. 

Statistical tables and graphs, 116. 

Statistics, 116, 121. 



117, 126, 148, 154, 
90, 96; steps in 



Tabular method 
155, 158. 

Tangent of an angle. 

finding, 96. 
Terms, like and unlike, 174. 
Translating word-statements, 12. 
Trinomial square, 266. 

Unknown distances, how to find, 54. 

Variables, 152, 319-327. 
Varying quantities, 319. 

Word problems, how solved, 33. 
Work and rate problems, 313. 



